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PROCEEDINGS 



OF THE 



Camkitrjje |pjtl0&0pjrird Sfltblj) 



VOLUME VII. 



(Eambritige : 

PRINTED BY C. J. CLAY, M.A. AND SONS, 
AT THE UNIVERSITY PRESS. 



PROCEEDINGS 



OF THE 



CAMBRIDGE PHILOSOPHICAL 
SOCIETY. 



VOLUME VII. 
October 28, 1889— May 30, 1892. 



QDambrfrge : 

PRINTED AT THE UNIVERSITY PRESS, 

AND SOLD BY 

DEIGHTON, BELL AND CO. AND MACMILLAN AND CO. CAMBRIDGE 
BELL AND SONS, LONDON. 

1892 



CONTENTS, 



VOL. VII. 



October 28, 1889. page 



By Pro- 



The President's Address ....... 

On Newton's Description of Orbits. By Dr C. Taylor . 
On impulsive Stress in Shafting, and on Kepeated Loading, 
fessor Karl Pearson ....... 

On Liquid Jets and the Vena Contracta. By H. J. Sharpe 

November 11, 1889. 

On the Varieties and Geographical Distribution of the common Dog- 
Whelk {Purpura lapillus L.). By A. H. Cooke . . . . 13 

On the Increase in Thickness of the Stem of the Cucurbitacece. By 

M. C. Potter . 14 

On the Spinning Apparatus of Geometric Spiders. By C. Warburton . 1 6 

On a new Result of the Injection of Ferments. By E. H. Hankin . 16 

November 25, 1889. 

On the Relation between Viscosity and Conductivity of Electrolytes. 

By W. N. Shaw 21 

On the Finite Deformation of a Thin Elastic Plate. By A. E. H. Love 31 

On a Solution of the Equations for the Equilibrium of Elastic Solids 
having an Axis of Material Symmetry, and its application to 
Rotating Spheroids. By C. Chree . . . . . . .31 

On the Concomitants of Three Ternary Quadrics. By H. F. Baker . 32 




Contents. 



January 27, 1890. 



On Non-Euclidian Geometry. By Professor Cayley .... 35 
On a Scheme of the Simultaneous Motions of a system of Rigidly con- 
nected Points, and the Curvatures of their Trajectories.- By J. 
Larmor 36 



February 10, 1890. 

On the perceptions and modes of feeding of fishes. By W. Bateson . 42 

Notes on Lomatophloios macrolepidotus {Goldg.). By A. C. Seward . 43 
On the origin of the embryos in the ovicells of Cyclostomatous Polyzoa. 

By S. F. Harmer 48 



February 24, 1890. 

On vehicles vised by the old Masters in Painting. Part I. A. P. Laurie 48 
On the Action of the Copper Zinc Couple on dilute solutions of Nitrates 

and Nitrites, NaHO and KHO being absent. By James Monckman 52 

On certain Points specially related to Families of Curves. By J. Brill . 57 



March 10, 1890. 

On the germination of Acacia sphaerocephala. By W. Gardiner . . 65 
Additional note on the thickening of the stem in the Cucurbitacece. By 

M. C. Potter 65 

Note on the action of Bennin and Fibrin-ferment. By A. S. Lea and 

W. L. Dickinson . . ... . . . . . .67 

On some Skulls of Egyptian Mummied Cats. By W. Bateson . . 68 



April 28, 1890. 

On the series in which the exponents of the powers are the pentagonal 

numbers. By Dr Glaisher 69 

On the Influence of Electrification on Ripples. By J. Larmor . . 69 
On Sir William Thomson's estimate of the Rigidity of the Earth. By 

A. E. H. Love . . . .72 



Contents. vii 

May 12, 1890. page 

On the action of Nicotin upon the Fresh-water Crayfish. By J. N. 

Langley 75 

On a new species of Phymosoma. Arthur E. Shipley .... 77 
On the action of the Papillary Muscles of the Heart. By J. George 

Adami 78 

On some living specimens of a Land-Planarian found in Cambridge. 

By S. F. Harmer 83 

May 26, 1890. 

On Solution and Crystallization. III. Rhombohedral and Hexagonal 

Crystals. By Professor Liveing 84 

On the Curvature of Prismatic Images, and on Amici's Prism Telescope. 

By J. Larmor 85 

On some theorems connected with Bicircular Quartics. By R. Lachlan 87 

October 27, 1890. 
On some Compound Vibrating Systems. By C. Chree .... 94 



November 10, 1890. 

Note on the principle upon which Fahrenheit constructed his Therrno- 

metrical Scale. By Professor Arthur Gamgee . . . . .95 
On Variations in the Floral Symmetry of certain Flowers having 

Irregular Corollas. By William Bateson and Miss Anna Bateson . 96 
On the nature of the relation between the size of certain animals and 

the size and number of their sense-organs. By H. H. Brindley . 96 
On the Oviposition of Agelena labyrinthica. By C. Warburton . . 97 
Supplementary list of spiders taken in the neighbourhood of Cambridge. 

By C. Warburton 98 



November 24, 1890. 

On the beats in the vibrations of a revolving cylinder or bell. By 

G. H. Bryan 101 

On Liquid Jets. By H. J. Sharpe Ill 

Note on the Application of Quaternions to the Discussion of Laplace's 

Equation. By J. Brill 120 

On a simple model to illustrate certain facts in Astronomy, with a view 

to Navigation. By A. Sheridan Lea 125 

Note on a paper relating to the Theory of Functions. By W. Burnside 126 



Vlll 



Contents. 



January 26, 1891. page 

On the Electric Discharge through rarefied gases without electrodes. 

By Professor J. J. Thomson . 131 

On the Laws of the Diffraction at Caustic Surfaces. By J. Larmor . 131 
On the effect of Temperature on the Conductivity of Solutions of 

Sulphuric Acid. By Miss H. G. Klaassen . . . . .137 



February 9, 1891. 

On Rectipetality and on a modification of the Klinostat. By Miss 

Dorothea F. M. Pertz and Mr Francis Darwin . . . .141 

On the Occurrence of Bipalium Ketvense, Moseley, in a new Locality; 

with a Note upon the Urticating Organs. By A. E. Shipley . .142 

On the Medusae of Millepora and their relations to the medusiform 

gonophores of the ffydroniedusce. By S. J. Hickson . . .147 

On the Development of the Oviduct in the Frog. By E. W. MacBride . 148 



February 23, 1891. 

On Tidal Prediction — a general account of the theory and methods in 

use and the accuracy attained. By Professor G. H. Darwin . .151 

On Quaternion Functions, with especial Reference to the Discussion 

of Laplace's Equation. By J. Brill 151 



March 9, 1891. 

On the disturbances of the body temperature of the fowl which follow 

total extirpation of the fore-brain. By J. George Adami . .156 

On the nature of Supernumerary Appendages in Insects. By W. 

Bateson 159 

On the Orientation of Sacculina. By Theo. T. Groom . . . .160 

On some experiments on blood-clotting. By Albert S. Griinbaum . .163 



May 4, 1891. 

On the most general type of electrical waves in dielectric media that is 

consistent with ascertained laws. By J. Larmor . . . .164 

On a mechanical representation of a vibrating electrical system, and its 

radiation. By J. Larmor 165 

On the Theory of Discontinuous Fluid Motions in two dimensions. 

By A. E. H. Love . . .175 

On thin rotating isotropic disks. By C. Chree 201 



Contents. 



May 18, 1891. page 

On Parasitic Mollusca. By A. H. Cooke 215 

Exhibition of models of double supernumerary appendages in Insects : 
also of a mechanical method of demonstrating the system upon 
which the Symmetry of such appendages is usually arranged. By 

W. Bateson 219 

On the nature of the excretory processes in Marine Polyzoa. By S. F. 

Harmer 219 



June 1, 1891. 

On the part of the parallactic class of inequalities in the moon's motion, 
which is a function of the ratio of the mean motions of the sun 

and moon. By Ernest W. Brown . 220 

On Pascal's Hexagram. By H. W. Richmond 221 

On a Linkage for describing Lemniscates and other Inverses of Conic 

Sections. By R. S. Cole 222 

On some experiments on liquid electrodes in vacuum tubes. By C. Chree 222 
Note on a problem in the Linear Conduction of Heat. By G. H. Bryan 246 



October 26, 1891. 

On the Absorption of Energy by the Secondary of a Transformer. 

By Professor J. J. Thomson 249 

On an experiment of Sir Humphry Davy's. By G. F. C. Searle . . 250 

Some notes on Clark's Cells. By R. T. Glazebrook and S. Skinner . 250 
Illustrations of a Method of Measuring Ionic Velocities. By W. C. D. 

Whetham 250 

On Gold-Tin Alloys. By A. P. Laurie 250 



November 9, 1891. 

Note on a Peripatus from Natal. By A. Sedgwick .... 250 

On "Variations in the Colour of Cocoons (Saturnia carpini and Eriogaster 

lanestris), with reference to recent theories of Protective Coloration. 

By W. Bateson 251 

Exhibition of Phylloxera vastatrix. By A. E. Shipley .... 252 
The Digestive Processes of Ammoccetes. By Miss R. Alcock . . . 252 
On the Reaction of certain Cell-Granules with Methylene-Blue. By 

W. B. Hardy 256 



Contents. 



November 23, 1891. page 

The Self-induction of Two Parallel Conductors. By H. M. Macdonald . 259 

The Effect of Flaws on the Strength of Materials. By J. Larmor . 262 
The Contact Relations of certain Systems of Circles and Conies. By 

W. McF. Orr 262 

On Liquid Jets \nider Gravity. By H. J. Sharpe 264 

Theory of Contact- and Thermo-Electricity. By J. Parker . . . 269 



February 8, 1892. 
On Long Rotating Circular Cylinders. By C. Chz-ee .... 283 

February 22, 1892. 

Some preliminary notes on the anatomy and habits of Alcyonium 

digitatum. By Sydney J. Hickson ....... 305 

On the action of Lymph in producing Intravascular Clotting. By Dr 

Lewis Shore 308 

On the fever produced by the Injection of sterilized Vibrio Metsehnikovi 

cultures into rabbits. By E. H. Hankin and A. A. Kanthack . 311 

Note on the Method of Fertilisation in Ixora. By J. C. Willis . .313 

March 7, 1892. 

Some Experiments on Electric Discharge. By Pz'ofessor Thomson . 314 
On the perturbation of a comet in the neighbourhood of a planet. By 

G. H. Darwin 314 

On the change of zero of Thermometers. By C. T. Heycock . . 319 
On the Elasticity of Cubic Crystals. By A. E. H. Love . . .319 
Changes in the dimensions of Elastic Solids due to given systems of 

forces. By C. Chree 319 

On the law of distribution of velocities in a system of moving molecules. 

By A. H. Leahy 322 



May 2, 1892. 

The application of the Spherometer to Surfaces which are not Spherical. 

By J. Larmor 327 



Contents. 



May 16, 1892. page 

Recent advances in Astronomy with Photographic Illustrations. By 

H. F. Newall 330 

On the pressure at which the electric strength of a gas is a minimum. 

By Professor Thomson 330 

On a compound magnetometer for testing the magnetic properties of 

iron and steel. By G. F. C. Searle 330 



May 30, 1892. 

On the hypothesis of a liquid condition of the Earth's interior con- 
sidered in connexion with Professor Darwin's theory of the genesis 

of the Moon. By 0. Fisher 335 

On Gynodicecism in the Labiatae. By J. C. Willis .... 348 

On the Steady Motion and Stability of Dynamical Systems. By 

A. B. Basset 351 



ERRATUM. 



344, line 14, for jf read 



172 772 mg ' 



PEOCEEDINGS 



OF THE 



Camhrifyje ^^ik^ap^ka;! Bututy. 



October 28, 1889. 

ANNUAL GENERAL MEETING. 

Mr J. W. Clark, President, in the Chair. 

The following Fellows were elected Officers and new Members 
of Council for the ensuing year : 

President: 
Mr J. W. Clark. 

Vice-Presidents : 
Dr Routh, Prof. Babington, Prof. Liveing. 

Treasurer: 
Mr Glazebrook. 

Secretaries: 
Mr Larmor, Mr Harmer, Mr Forsyth. 

New Members of Council: 

Prof. Cayley, Prof. Darwin, Prof. Lewis, Dr Gaskell. 

VOL. VII. pt. i. 1 



2 President's Address. [Oct, 28, 

The names of the benefactors of the Society were recited by 
the Secretary. 

On the motion of the President, seconded by the Treasurer, 
it was resolved : — That during the ensuing session the meeting of 
the Council of the Society be fixed for 4 o'clock in the afternoon; 
that the meeting of the Society be fixed for 4.30, when tea will 
be provided, and that the formal business of the meeting be taken 
at 5 o'clock. 

The President delivered the following address : 

I rise, according to custom, to say a few words at what would, 
under ordinary circumstances, be the close of my first year 
of office as your President. It happened, however, that in con- 
sequence of the lamented death of Mr Coutts Trotter I was 
elected on Monday, 30 January, 1888, instead of at the usual time 
in November. I have therefore had the honour of holding the 
office for a year and nine months. 

During that period the Society has pursued the even tenour 
of its way, undisturbed by revolutions or dissensions, but at the 
same time giving signs of vigorous and healthy life. It is evident 
that we can no longer expect that a majority of those elected to 
College Fellowships in this University will seek the further dis- 
tinction of our Fellowship, almost as a matter of course, as they 
used to do; but it may still be a subject of congratulation that 
our annual recruits make up in quality for defects in quantity; 
and, if I am not too sanguine, I think that the ancient popularity 
of our Society may, at any rate to some extent, be revived. 

Our meetings have been well attended; but it seems to me 
that it might be possible to render them still more attractive, and 
so to make them fortnightly gatherings of those who, being in- 
terested in scientific pursuits, are anxious to meet persons of 
tastes similar to their own. - With this object in view a proposal 
will be submitted to you for changing the hour of meeting. You 
remember that in 1881-82 the hour was changed from eight in 
the evening to three in the afternoon. That change was received 
with approbation; but since then it has been represented to the 
Council that a further change to a somewhat later hour in the 
afternoon would now be more convenient. It is now, therefore, 
proposed to meet at half-past four; and further, to offer, before 
the meeting, that refreshment which, in all circles, whether rich 
or poor — scientific, literary, or social — offers a resting-place be- 
tween lunch and dinner — or between dinner and supper — namely, 
tea. It was remarked long ago that " great events from little 
causes spring;" so let us hope that this innocent beverage, which 
has not, as yet, fallen under the ban of any school of reformers, 



1889.] President's Address. 3 

may operate as a charm to bring our members together, and pro- 
mote the ends we have in view. 

There is another matter to which I would briefly allude — our 
Library. The importance of having a central scientific Library 
in these buildings, in addition to departmental libraries, has long 
been recognised in principle; and I cannot but think that when 
the practical usefulness of it becomes more widely known, members 
of the University will gladly support it by becoming Fellows 
of the Society. At present the number of students who use it 
steadily increases ; but I am sorry to say that the power of the 
Society to support it is diminishing. In this year it became 
necessary to ask the Museums and Lecture Rooms Syndicate to 
defray the cost of certain periodicals which had heretofore been 
borne by the Society. This the Syndicate agreed to do ; but I 
would remark that, while they fully recognise the value of the 
Library, and are willing to spend money liberally upon it, the 
funds at their disposal are by no means large, having regard to 
what they have to do with them. Meanwhile the Library is 
deficient in numerous works which are being continually asked 
for, and which neither the Society nor the Syndicate are rich 
enough to purchase. 

The smallness of our funds operates to our disadvantage in 
another way — we are unable to illustrate our publications as fully 
as is desirable. You will, I am sure, give credit to those in charge 
of our Proceedings for the pains they bestow upon them, and for 
the praiseworthy rapidity with which they are circulated. A 
number of the Proceedings completing Vol. vi. is now on the 
table ; and a part of the Transactions, completing Vol. xiv. is 
now ready. 

I beg to return my most cordial thanks to the Council and the 
officers of the Society for the assistance they have rendered to 
me personally; and I am sure that you will recognise the admirable 
zeal with which they have discharged their duties to the Society. 
The Treasurer, Mr Glazebrook, should be specially congratulated 
on the success which has attended his efforts to obtain payment 
of arrears of subscriptions due to the Society. 

Lastly, it is my duty to record the names of those Fellows of 
the Society whom we have lost by death during the past year. I 
will take them in the order of seniority. 

The Rev. Richard Okes, D.D., Provost of King's College. 

The Rev. Benjamin Hall Kennedy, D.D., Regius Professor of 
Greek. 

William Henry Drosier, M.D., Fellow of Gonville and Caius 
College. 

The Rev. Churchill Babington, D.D., formerly Fellow of St 
John's College. 

1—2 



4 Br 0. Taylor, On Newton's description of Orbits. [Oct. 28, 

John Reynolds Vaizey, M.A., Peterhouse. 

Besides these, we have lost one honorary member: 

James Prescott Joule, F.R.S. 

It had been my intention to attempt a short biography of 
each of these; but I found that I had neither time nor materials 
to perform such a task efficiently. Moreover, they are for the 
most part too well known, and too intimately connected with 
this University, to need such commendation. It would, however, 
be a personal satisfaction to myself to remind you that in Mr 
Vaizey — who died from the results of an accident at the beginning 
of this year — the Biological School has lost an energetic worker, 
whose usefulness as a teacher had been already recognised, and 
who, had he lived, would probably have risen to eminence in his 
own special science, Botany. 

The following Communications were made to the Society: 

(1) On Newton s description of orbits. By Charles Taylor, 
D.D., Master of St John's College. 

The Master drew attention to the fact that the problem of 
constructing a Conic Section to satisfy given conditions has been 
treated incidentally with great power and considerable complete- 
ness in the Principia. A comparison was made between the 
methods of Newton and the more modern methods: and some 
improvements were suggested. The way in which Newton passes 
from cases of real intersection of lines with conies to cases in 
which real points of intersection do not exist, strongly suggests 
the question whether he had possession of the idea of imaginary 
points, which is usually ascribed to a much later period. 

(2) On impulsive stress in shafting, and on repeated loading. 
By Prof. Karl Pearson, University College, London. 

(3) On Liquid Jets and the Vena Contracta. By H. J. Sharpe, 
M.A., St John's College. 

1. When liquid flows out of a vessel through an orifice, a 
liquid particle in contact with the vessel describes an ordinary 
stream-line as long as the particle is within the vessel, but the 
moment it escapes through the orifice, this stream-line suddenly 
becomes also a line of constant velocity, if no force acts on the 
liquid. In the solutions presently to be given, which are capable 
of infinite variety, the coincidence between the outer stream-line 
of the jet and a line of constant velocity is not (as in Kirchhoff's 
solutions) mathematically perfect, but (even near the orifice, 
where it is most imperfect) can be made, as will be seen from 
examples, very close, and as we pass along the jet. becomes very 



1889.] 



Mr H. J. Sharpe, On Liquid Jets, <lx. 



rapidly nearly perfect. If the position of the orifice (subject 
however to the limitation that the coefficient of contractoin must 
be >^) be arbitrarily chosen, it will be seen that it can be so 
chosen as to make the coincidence as close as may be desired. If 
however, which seems the more correct course, it be chosen from 
a consideration presently to be given (Art. 5) it is not easy, I 
think, to say for certain whether the above-named coincidence 
can be made as close as may be desired. It is an interesting ques- 
tion which remains for solution. The motion is in two dimensions 
and everywhere finite. The species of vessel for which solutions 
are obtained may be described generally as a canal, whose sides 
up to a certain point are straight and then turn off abruptly at 
right angles into a curve towards an orifice, at the axis of the 
canal, on each side of which the fluid motions are symmetrical*. 
The jet ultimately approaches an asymptote parallel to its axis. 
The ratio of the breadth of the vessel to the ultimate breadth 
of the jet can be made anything we like, but we shall always 
suppose it an even integer. 

2. Liquid is supposed to be flowing from right to left, roughly 
speaking parallel to the axis of x, which is taken as the axis of 
the vessel and jet. The stream-line AFBH is taken for the 
boundary of the vessel. Fig. 1 may be taken as the type of the 



Fig. 1. 





y 

E 




F 




A 


^— -— - 


li 

D 




-t 























general case. OE is always ir, and OB, OD submultiples of it. 
In figures of which fig. 2 is a type B and D coincide. At present 



* It will be seen, however (Art. 6), by an obvious extension of the method, that 
it is not confined to canals having this peculiarity, but is applicable also to canals 
having flowing curvilinear boundaries. 



Mr H. J. Sharpe, On Liquid Jets 



[Oct. 28, 



the only limitation we shall put upon the position of the orifice 
H is that its ordinate must not exceed twice 01). 




x> 



3. Different analytical expressions, containing an arbitrary 
number of arbitrary constants, will be assumed for the velocities 
on either side of Oy, but it will be shewn that they can be so 
chosen as to make the velocities on either side of Oy continuous, 
and leave any required number of arbitrary constants to satisfy 
conditions now to be given. It will be shewn that in all cases 
the equation to BHG can be expressed in the form (putting, for 
brevity, z for e*), 



y = a — c l z sin y — \ c. 2 z 2 sin 2y — &c. 
and the velocities on the left of Oy in the form 

— -r = 1 + c t z cos y + c 2 z 2 cos 2y + &c. ; 

v 

-j = c x z sin y + c 2 z 2 sin 2y + &c, 



■(1), 



(2), 



where of course a. is the ultimate value of Cos'. We are concerned 
only with the velocities at all points along HG, so that x and y 
in (1) are the same as x and y in (2). As z is less than 1 we can 
solve (1) so as to express y in a series of ascending powers of z. 
We shall have therefore at every point of HG to the second order 
of approximation 

y = a — c 1 z sin a + &c (3), 



A 



5 (u 2 + v 2 ) = 1 + 2CjZ cos a + &c. 



.(4). 



1889.] and the Vena Contractu. 7 

If a = 7r/2 the coefficient of z vanishes, and it will be shewn that 
the remaining disposable constants can be so chosen as to make 
the coefficients of z'\ z z , &c. (any desired number of them) also 
vanish. 

Next suppose that c 1 — 0, then to the third order of approxi- 
mation 

JL (i? + v 1 ) = 1 + 2c 2 z 2 cos 2<x + &c. 

If a = 7r/4 the coefficient of z 2 vanishes, and if there are con- 
stants enough, the coefficients of z z , z*, &c. could be made to 
vanish. And generally if c^c^... c m _ 1 vanish, then to the (m + l)th 
order of approximation 

_L (y + v *) = i + 2c m z Kl cos ma (5), 

and if a = ir/2m the coefficient of z m vanishes, and the coefficients 
of z m+1 &c. could be made to vanish. 

We have supposed at first, for simplicity, all the powers of z 
in (I) to exist, but in every case we shall find that many are 
wanting. For instance, after the first few terms, only even mul- 
tiples of y may occur — or again multiples of 6y. Again, even if 
(1) is complete, yet it will be found that if a — tt/2, (3) will 
consist only of odd, and (4) of even powers of z. This considera- 
tion not only simplifies the work, but also enables us with greater 
ease to make (u 2 + v 2 ) constant to a higher degree of approxi- 
mation. 

4. We shall now shew how to make the velocities on either 
side of Oy continuous. We shall take the case to which fig. 2 
refers, where OB is irj2 and w T here the ultimate value of Cx' is 
OB. I give this case, not only because it is one of the first I 
solved, but because I believe it exhibits the method under the 
greatest disadvantages, and yet it will be found that the approxi- 
mation obtained is very close. 

We will take for the stream function y}r, and the velocities on 
the left of Oy, 

^ = — u = a r e x cos y + a 3 e BX cos Sy + Xa 2n e 2nx cos 2ny + A 



dy 

-—- = v = a x e x sin y + a 3 e 3x sin Sy + 2<2 2M e 2 ' !X sin 2ny 



(6), 



S indicates summation for all integral powers of n from 1 to oo . 
Only two odd multiples of y appear in the above, but it must be 
distinctly understood (in fact it is the characteristic feature of the 
present method) that any number of any odd multiples of y could 



8 Mr H. J. Sharpe, On Liquid Jets [Oct. 28, 

be similarly used, and each assumption would give a distinct case. 
The same also must be understood for the stream function and 
velocities presently to be assumed on the right of Oy. 
Then the equation to BUG is 

a L e x sin y + ±a./ x sin 3^ + 2 -^ i nx sin 2ny 

, CL a A.7T /H _ N 

+ Ay = a 1 -^ + ~. ..{!). 

Since the ultimate breadth of the jet is supposed to be 7r/2, we 
must have 

«-x-K=o («)• 

It will be convenient to replace a v a 3 each by two new quan- 
tities, such that 

a^^ + A^ a 3 = cx 3 +A 3 (0). 

Then when x = 0, we have at every point of OB, on the left of Oy 

-u= (a, +A 1 )cosy+ (a 8 + A 3 ) cos Sy + ta 2n cos 2ny + A ) 
v = (a l + AJ sin y + (a 3 + A 3 ) sin oy + %a 2n sin 2ny ) 

On the right of Oy assume for the velocities 

dy \ 

— -^ = —u = b x e~ x cos y + b 3 e Zx cos Sy + 2,b 2n e~ 2nx cos 2ny + B 

I (11) - 

_ _* = v = - b x e x sin y — b 3 e 3x sin Sy - %b 2n e 2nx sin 2ny 

It will be convenient to replace b v b 3 each by two new quan- 
tities, such that 

b 1 ^cc 1 -A 1 , b 3 = cc 3 -A s (12). 

Then when x = 0, we have at every point of OB, on the right 
oiOy, 

-u = (a 1 - AJ cos y + (cc 3 -A 3 ) cos Sy + tb 2n cos 2ny + B~\ 

v = -(a l -A 1 )smy-(cc 3 -A 3 )smSy-Zb 2n sm2ny J 

By Fourier's theorem, suppose we have, from y = to 7r/2, 

2A 1 cosy+ 2A 3 cosSy=p + 2p 2n cos 2ny (14). 

This will be true at both limits. Then the first equation of (10) 
Avill be identical with the first equation of (13) if 

* 2n -K+p, n = o ( 15 )> 

A-B+p = (16). 



1889.] and the Vena Contracta. 9 

Again, by Fourier's theorem, suppose we have from y = to 
tt/2 

2^8111 2/ + 2a 3 sin Sy = 1q 2n sin2ny (17). 

This will be true at both limits if 

«t-«. = (18). 

Then the second equation of (10) will be identical with the second 
equation of (13) if 

%>+&*» + &„ = <> (19). 

Then if the constants satisfy equations (15), (16), (18), (19) the 
motions on the left and right of OB will be continuous. 
From (11) the equation to AFB is 

b^ x sin y + i b 3 6~ 3x sin Sy + X -^ e -2 "* sin 2ny 



+ %=& 1 -| + ^...(20). 



5. We will now suppose that 



y-|-£ < 21 > 

When (21) is fulfilled, the stream-line AFB will consist of an 
infinite straight line AF, whose ordinate is it, and a curved portion 
FB. The peculiarity at F will be presently explained. It must 
be carefully observed that AF is not an asymptote. 
It will be found that we get the following relations 

o 

a^^ + A,, ^ = -^-(17^ + 16^), 

a 3 = 8^+ SA V a 4 = - JL. (61a, + 644,), 
a, n 2 L ( 4rc 4rc ± 12\ / ± 2 _ 18 \] 

&! = .«! -A, 6, = ^^ + 16^), 

ft. — e^-S^, 6 4 = I p^(29« 1 + 644 1 ), 

. J 6a, „ 8a, , s 

^=^- x , B = ^ (22). 

It will be noticed that for moderately large values of n, a 
and b tn ultimately vary as l/?i* so that all the series employed 



10 



Mr H. J. Sharpe, On Liquid Jets 



[Oct. 28, 



are convergent even for points very near Oy. I believe this 
property is common to all the solutions obtained by the present 
method. It is interesting to notice that unless we assumed the 
equation (18) to hold, we could not establish, the obvious relation 
A=2B. 

6. We will now make the curve HG as far as possible 
identical with a line of constant velocity. It will be convenient 
to put for shortness e x = z, 

aJA = c t , aJA = c 2 , aJA = c 3 , &c. 
Then (7) can be written 



y = 9 — c x z sin y — \c 2 z 2 sin 2y — &c. 

and from (6), 

— j = 1 + c x z cos y + c 2 z 2 cos 2y + &c. 
A 

v 
—r = c x z sin y + c 2 2 2 sin 2y + &c. 

In (23) expressing y in terms of z, we have 

V = I - G i z + (K 3 - c i c 2 + i c 8 ) z 3 + &c. 



.(23), 



.(24). 



(25). 



It will be found that this series consists only of odd powers of z, 
the simplification arising from our having taken 7r/2 for the ulti- 
mate breadth of the jet. If we now substitute the above value 
of y in (24) and form the value of u 2 + v 2 , we shall get 

25(m- + 0=1 + (3c*-2c 2 )z* 

+ (- f c/ + 8 Cl 2 c 2 + c 2 2 - ^ Cl c 3 + 2c 4 ) ** + &c. . .(26), 

the series consisting only of even powers of z. 

If we want to carry the approximation to the fourth order, we 
shall have to cause the coefficient of z l here to vanish. This will 
give us for determining the ratio of A x to a t the equation 

Solving this equation we get 

4,/^ = -1-0635 or -4-0127 nearly (27). 

Either of these gives a solution. If we take the first, we see 
from the equations (22) at the end of Art. 4, that all the quan- 
tities a t> a 2 , a 3 , a 4 are small, therefore the curve BHG clings all 
along very closely to its asymptote. 



1889.] and the Vena Contracta. 11 

From equations (6) or (11) we shall get for the direction of 
the fluid motion at B, 

t'_37T/ A 

u ~ 1 V + ^ 

For the first solution this becomes — '06, and for the second — 2"84. 

If we wished to carry the approximation to the sixth order, we 
should have to introduce terms with the sine and cosine of 5y in 
(6) and (11) and make the coefficient of z* in (26) vanish. This 
solution would involve two arbitrary constants. 

By making in addition the coefficient of z® in (26) vanish we 
could carry the approximation to the 8th order, and the solution 
would involve one arbitrary constant. 

7. We proceed now to discuss the solutions obtained. And 
first for the point F, where there is a peculiarity which was 
pointed out to me by Sir George Stokes. 

From (20) the equation to AFB can be written (putting for 
shortness z for e - *), 

B (y — ir) — h x z sin {y — ir) + \b z z z sin 3 (y — ir) 

-$ b f~sm2n(y-Tr) (28). 

Dividing out by (y — ir) and then putting y = it, we get for deter- 
mining the abscissa of F, 

B^b.z + bZ-lb^ (29). 

For the first solution in (27) this gives us z = *58, from which we 
get x — 23/43 nearly. 

Further it will be seen from (28) that for points near F, z 
does not vary, for on account of the values of z and b (see 
end of Art. 4) the series converges pretty rapidly. Therefore 
the stream-line turns sharply at right angles at F. This curious 
result may be further corroborated simply by comparing the first 
equation of (11) and (29) from which we at once get u = at F. 
It may be noticed that we are not compelled to make the ultimate 
ordinate of the stream-line on the right of Oy equal to ir. For 
instance, in the present case, we arbitrarily assumed (21) to hold. 
If we did not assume this, the outer stream-line on the right of 
Oy would have no sharp corner. The ultimate ordinate however, 
whatever it is, must not exceed it, otherwise the motion would be 
discontinuous. 

8. Kirchhoff has shewn (Lamb's Motion of Fluids, Art. 96) 
that at such a point as H (where the motion of the fluid having 



12 Mr H. J. Sharpe, On Liquid Jets, &c. [Oct. 28, 

been restricted suddenly becomes free) the radius of curvature 
should be zero in the true solution. In the present approximate 
solutions we cannot of course expect this to hold. But I am 
inclined to believe that in all solutions obtained by the present 
method there will be found on the line BHC (near where we 
might expect the orifice to be) a point where the radius of curva- 
ture is a minimum, and here I propose to place the point H. At 
any rate there is such a point in the case of the first solution in 
(27), as we proceed to shew. 

From (25) putting in it c 2 =|c 1 2 and from (8) c 3 = 3a a , the 
equation to BHC is 

2 / = !-c 1 * + (-c 1 3 + c > 3 + &c (30). 

From (27), &c. 

_a A = Vj-A_ _?7r 0635 

It is evident therefore that c l is small and that (30) may be ap- 
proximately written 

y=\ -c 1 (^-/)+&c. 

As dyjdx is always small all along BHC, we may get sufficiently 
near the point required by finding the point where d 2 y/dx 2 is a 
maximum. We see at once that this is got from the equation 
z 2 is equal to 1/27 which gives x=— 71/43 nearly for the abscissa 
of H. We can readily see that at H the curve is convex to the 
axis and that a maximum value of d 2 y/dx 2 has been obtained, 
also that the coefficient of contraction is "99545. 

9. It will now be interesting to calculate the limits of error 
in the velocity at the point H. From (26) it can be shewn that 
the coefficient of z 4, reduces to 

-\f-c l i -26c*+2c i . 

This is equal to '1654 nearly. Therefore at the point H only 
•000011 of the velocity is variable. Of course as we pass to the 
left of H this small proportion rapidly diminishes. 

10. If we examine equation (20) we shall find that the curve 
FGB cuts the line y — tt/2 in a point G whose abscissa is about 
3/43. Also if we imagine two points on the same curve whose 
ordinates are 47r/6 and 57r/6 we shall find that their respective 
abscissae are about 16/43 and 36/43. 



1889.] Mr A. H. Cooke, On the common Dog- Whelk. 13 

November 11, 1889. 

Dr Gaskell in the Chair. 
The following Communications were made: 

(1) On the Varieties and Geographical Distribution of the com- 
mon Dog-Whelk {Purpura lapillus L.). By A. H. Cooke, M.A., 
King's College. 

The author, while unable to advance any satisfactory theory 
to account for colour variation, held that variations otform largely 
depended upon the station occupied by the animal. Shells occur- 
ring in exposed situations (e.g. Land's End, Scilly Islands, coasts 
of N. Devon and Cornwall) were stunted, with a short spire and 
large mouth, the latter being developed in order to increase the 
power of adherence to the rock, and of resistance to wave force. 
Shells occurring in sheltered situations, estuaries, narrow straits, 
&c. where there was no severe wave force to encounter, were of 
great size, spire well developed, mouth small in proportion to 
area of shell. This view was illustrated by series of specimens 
collected at various points on the British coasts. 

With regard to the question of geographical distribution it was 
shewn th&tPurpura lapillus (a "north temperate" species) occurred 
on the East Asiatic coasts from Behring's Straits to Hakodadi 
(41°), on West European coasts from North Cape to Mogador 
(32°), not entering the Mediterranean, on East American coasts 
from Greenland to Newhaven (42°), and on West American coasts 
(assuming the identity of the West American Purpuras with 
lapillus) from Alaska to Margarita Bay (24°). Thus on the two 
western coasts it had a far more southern range than on the two 
eastern. The author regarded this fact as due to the direct 
influence of the surface temperature of the ocean. The mean 
annual temperature (taken from the Meteorological Society's 
charts) of the surface water at Hakodadi was 52°, with an extreme 
range of 25°; that of Mogador was 66°, extreme range only 8°; 
that of Newhaven was 52°, extreme range 30°; that of Margarita 
Bay 73°, extreme range only 5°. Violent changes of temperature 
were fatal to life, zones where such changes occurred acted as 
barriers to distribution; it was possible on the other hand for an 
organism to bear a gradual change from cold to extreme heat. 
On the western coasts of Europe and America the change from 
cold to heat was very gradual, hence the Purpura had been able 
to creep as far south as 32° in the one case and 24° in the other; 
while on the opposite eastern coasts, where the Atlantic and 
Pacific Gulf-streams caused a sudden change in the temperature 
of the surface-water, the species was barred back at a point many 
decrees further north. 



14 Mr M. C. Potter, On the increase of the thickness [Nov. 11, 

(2) On the increase in thickness of the stem of the Cucurbitacece. 
By M. C. Potter, M.A., St Peter's College. 

The Order Cucurhitacem consists for the most part of her- 
baceous plants climbing by means of tendrils; like many other 
climbing plants, the members of this order have an anomalous 
distribution of the fibro-vascular bundles in the stem; the bundles 
being arranged in two concentric rings, and each individual bundle 
being bicollateral, with phloem both on its external and internal 
sides. (Fig. 1.) 

The structure of these stems was first described by Hartig* 
and then by von Mohl*f" and has been the subject of investigation 
by various botanists, most of whom have confined their attention 
to the structure and contents of the sieve tubes. Bertrandj how- 
ever has described the manner in which a cambium between the 
xylem and both inner and outer phloem adds respectively both 
xylem and phloem to the bundle, whilst Petersen §, in his article 
on Bicollateral Bundles, gives a short account of the increase of 
the stem of Zehneria suavis; stating that there is no interfascicular 
cambium, but that while the bundles increase the cells of the 
medullary rays increase passively in a radial direction and finally 
divide. With this statement de Bary || agrees. Fischer If also 
says that there is no interfascicular cambium present whereby 
these stems can increase in thickness. The fact that all the 
investigated species of Cucurbitacem have been herbaceous ex- 
plains why the interfascicular cambium has hitherto not been 
described. 

Lately I have had an opportunity of investigating the woody 
stems of Cephalandra indica (Naud.), Trichosanthes villosa (Bl.) and 
T. anamalayana (Bedd.), and find that they increase by a well- 
marked interfascicular cambium. The stems of these plants agree 
with those of the other members of the order in possessing the 
two rings of bicollateral bundles (fig. 1), but differ in having no 
ring of sclerenchymatous tissue between the epidermis and vas- 
cular bundles. The structure of these species being similar in all 
respects it will only be necessary to describe the stem of Cepha- 
landra indica. 

Cephalandra indica, a climbing woody plant, reaches to a 
considerable height and its stem, which is perennial, attains to 
several inches in diameter. In its young stage the stem contains 

* Bot. Zeit. 1854. 

t Bot. Zeit. 1855. 

J " Theorie du faisceau," Bull. Sci. du departement du Nord, 1880. 

§ Eiigler Bot. Jahrb. vm. p. 374. 

|| Comparative Anatomy, English edition, p. 456. 

it TJntersuchungen iiber das Siebrohreii-system der Cucurbitaceen, p. 6. 



1889.] of the stem of the Cucurbitacece. 15 

ten fibro- vascular bundles in two concentric rings, the five inner 
bundles being larger than the five outer ones (fig. 1). The stem 
increases slowly for some time, because the cambium on the outside 
as well as on the inside of each bundle adds xylem and phloem 
to the bundle; the cells of the medullary rays increase radially 
and divide. Thus far Cephalandra is similar to the herbaceous 
members of the order. The latter, being annuals or at least not 
requiring a continual enlargement of the stem, have no provision 
for such increase, whilst Cephalandra being woody and perennial 
requires means whereby its stem can continue to grow, and there- 
fore by division of the cells of the medullary rays adjacent to the 
bundles and contiguous to the outer cambium cells there is formed 
an interfascicular cambium (fig. 2a). This interfascicular cambium 
soon stretches from bundle to bundle across the medullary rays 
(see dotted line fig. 1), and its development agrees with that of 
normal Dicotyledons. After the completion of the ring of inter- 
fascicular cambium, xylem is formed on the exterior of the existing 
xylem, and phloem on the interior of the outer phloem and secondary 
medullary rays are formed in the xylem in the normal manner 
(Plate II. fig. 4 mr z ). The stem therefore grows in the same 
manner as a normal Dicotyledon. The cambium which is placed 
on the inner side of the xylem ceases to produce new elements 
about the time the interfascicular cambium is formed, and finally 
disappears; so that the internal phloems are left at the centre of 
the stem and do not undergo further increase. 

The above description shews that the mode of increase of the 
stem of Cephalandra indica corresponds exactly with that of a 
normal Dicotyledon in which no intermediate bundles are formed, 
inasmuch as each possesses a cambium which forms on its inside 
xylem and medullary rays, and on its outside phloem and paren- 
chyma. But the stem of the former differs in the primary ar- 
rangement of the bundles in two rings and in having some phloem 
on the central side of the xylem. It would seem therefore that 
de Bary's* view that the two concentric rings of bundles in the 
stem behave as a single ring curving alternately outwards and 
inwards is correct. 



EXPLANATION OF PLATES I AND II. 

Fig. 1. Diagram shewing position and relative size of the fihro- 
vascular- bundles of the stem of Cephalandra indica. 
Ph = Phloem, Xy = Xylem, the wavy line indicating where 
the interfascicular cambium will be formed. 



Comparative Anatomy, English edition, p. 45G. 



16 Mr E. H. Hankin, On a new result [Nov. ,11, 

Fig. 2. Portion of stem of Cephalandra indica shewing two interior 
and one exterior fibro- vascular-bundle, a cells dividing to 
form the cambium. C centre of stem. Si sieve-tubes. P 
the periphery. 

Fig. 3. Portion of an older stem of Cephalandra indica. c centre of 
stem. Si sieve-tubes, cam cambium, ph position of interior 
phloem of two other fibro-vascular-bundles, mr l primary 
medullary ray, mr 2 secondary medullary ray. 

(3) On the Spinning Apparatus of Geometric Spiders. By C. 
Warburton, B.A., Christ's College. 

The structure of the external and internal spinning organs of 
the Epeiridod was described, and the special functions of the 
several distinct kinds of spinning glands investigated. 

The Ampullaceal glands were shewn to be the sources of the 
framework and radial lines of the geometric web. The Acinate 
and Piriform glands are those mainly used in binding up captured 
insects. 

A Spider's line is not composed of many strands interwoven 
or coalescent, as has been hitherto believed. It usually consists 
of two or four non-adherent threads, and when more are present 
they do not fuse, but remain distinct, although contiguous. 

The foundation line of the spiral consists of two strands only, 
not adhering on account of their own viscidity, but enveloped in 
a common viscid sheath which subsequently breaks up into bead- 
like globules, and which is probably furnished by the aggregate 
glands. 

(4) A new result of the injection of ferments. By E. H. Han- 
kin, B.A., St John's College (Junior George Henry Lewes Student). 

The following experiments were performed in Professor Koch's 
laboratory at the Hygienisches Institut, Berlin. Although the 
theoretical considerations that led me to perform these experi- 
ments are by no means proved by them, the results appear to be 
sufficiently interesting to publish in detail. 

Experiment 1. Three rabbits, Nos. 11, 12, and 13, were inocu- 
lated with virulent anthrax*. No. 13 served as control and died 
in 36 hours of typical anthrax. Nineteen hours after inoculation 
rabbits 11 and 12 were subjected to a further treatment. No. 11 
had two cubic centimetres of a '1 per cent, solution of trypsin 
injected into the lateral vein of its ear, and No. 12 had two cc. 
of a one per cent, solution of pepsin similarly injected -f*. 

* These rabbits were of medium size, No. 11 weighed 855 grammes. 

t The pepsin and trypsin employed were both obtained from Schering's Griine 
Apotheke, Chaussee Strass, Berlin. In my later experiments I employed some very 
pure pepsin which I owe to the great kindness of Dr Theodor Weyl. 



1889.] of the Injection of Ferments. 17 

Rabbit 12 died 66 hours after its inoculation with anthrax. 
The anthrax bacilli in the lung, spleen, and lymph gland near 
seat of inoculation, instead of appearing in the form of short 
rods characteristic of virulent anthrax, were arranged for the 
most part m long chains as is usual with attenuated virus. The 
chains consisted of as many as 12 and sometimes even more 
joints. Two mice were inoculated from the heart-blood of this 
rabbit, and died of anthrax after the rather unusually long period 
of 60 hours. No. 11 died only 13 days after its inoculation. It 
had slight diarrhoea for some days before its death. Its spleen 
contained but few bacilli arranged in chains generally of six or 
seven but sometimes of as many as fifteen joints. 

Three days after the inoculation of these rabbits another rabbit, 
22, was inoculated from the same culture. The next day three cc. 
of one per cent, pepsin was injected intravenously. The rabbit 
died six days afterwards and its spleen shewed very few bacilli 
all arranged in chains. 

Experiment 2. Three rabbits were inoculated with virulent 
anthrax. Two days afterwards only one was still alive. It was 
treated with four cc. of one per cent, trypsin injected into its ear 
vein. It died four days after its anthrax inoculation. The spleen 
contained numerous bacilli, which were for the most part arranged 
in chains, one of which contained as many as 24 joints. These 
chains shewed signs of degeneration, in that they stained very 
irregularly. In the same chain some joints were colourless while 
others were deeply tinted. The spleen of this rabbit was par- 
ticularly large, and a great many of the cells contained more 
than one nucleus, while other nuclei had a dotted aspect. Ap- 
parently these appearances indicate an increased rate of nuclear 
division*. 

Experiment 3. Professor Koch kindly inoculated for me five 
rabbits, Nos. 26 to 30, each with a large quantity of anthrax 
spores suspended in a normal salt solution. No. 29 served as 
control and died in 36 hours. I injected three to four cc. of one 
per cent, trypsin into the ear vein of Nos. 26, 27 and 28 directly 
afterwards. No. 26 died in the same time as the control. None 
of the bacilli were in very long chains, none of more than six 
joints were seen, while most of the bacilli were exceptionally 
short. This fact and also the way in which they were arranged 
on the slide suggested to me that the bacilli had at first grown in 
longer chains, but that at a later period (perhaps when the effect 
of the trypsin had passed away) they had broken up into separate 
segments. 

* My observations were all made on fresh preparations of the spleen pulp, to 
which some dilute aqueons solution of methyl blue was generally added. 

VOL. VII. PT. 1. 2 



18 Mr E. H. HanJcin, On a new result [Nov. 11, 

No. 27 died in about 50 hours. The bacilli in the spleen 
were rather longer than usual. 

No. 28 died in 21| hours. Unfortunately I have mislaid my 
notes of its post mortem appearances. 

No. 30 had four cc. of one per cent, trypsin injected intra- 
venously the day after it was inoculated with anthrax. It died 
in 36 hours and the spleen shewed the bacilli arranged in long 
chains. 

Experiment 4. Five rabbits, Nos. 34 to 38, were as before in- 
oculated by Professor Koch with anthrax spores. Directly after- 
wards I administered intravenously from one to 3^- cc. of 05 per 
cent, solution of trypsin. 

No. 38 was control and died between 50 and 60 hours after 
inoculation. 

No. 34 weighed 1500 grammes and had one cc. of the above 
solution of trypsin. It died 51 hours after inoculation. Some of 
the bacilli were isolated, but many were in chains of more than 
12 members. 

No. 35 weighed 1507 grammes and had 3^ cc. of the "05 per 
cent, trypsin solution. It died in 60 hours, and the bacilli were 
arranged in chains. No. 36 died after 36 hours*. The spleen 
contained very few bacilli and of these some w r ere arranged in 
chains. 

No. 37 weighed 1865 grammes. It had 2| cc. of the above 
solution of trypsin. It was very ill for some days, but at last 
recovered and is now alive and well nearly three months after the 
operation. The temperature record of this unfortunately unique 
case is very interesting. The day after inoculation the temperature 
was 37°"4 Centigrade, i.e. 2*4 degrees below the normal tempera- 
ture of a rabbit. It remained at approximately this low figure for 
some days, shewing a very gradual rise, and only on the sixth 
day after inoculation had it reached 38°. From this point it 
rapidly rose till on the 11th day after inoculation it was 40°'l. 
On the 12th day it stood at 40 o, 05, when observation of its tem- 
perature was discontinued. Another interesting point about the 
case was the appearance of pus at the seat of inoculation. On 
the 8th day after the experiment began, a small hard tumour 
about half-an-inch in diameter was found at the seat of inocu- 
lation. On the 13th day a second larger tumour appeared in 
front of the former. This gradually increased in size and was 
found to contain caseating pus. About a week later, no further 
increase in size could be noted. The animal appeared to be ema- 



* I noticed while this rabbit was being inoculated that it had a cough. This 
point is worth noting, as I have usually observed that a lung disorder increases 
susceptibility to anthrax. 



1889.] of the Injection of Ferments. 19 

ciated. At the present time it appears to be strong and fat, and 
only a trace of the swellings can be seen. 

Experiment 5. Three rabbits, Nos. 40, 41 and 42, were in- 
oculated with virulent anthrax spores. No. 42 served as control 
and died within 60 hours. Its spleen contained very few bacilli 
which were never arranged in chains of more than six joints. 
No. 40 received five cc. of one per cent, pepsin solution the day 
after inoculation with anthrax. It died after about 60 hours. 
The bacilli in the spleen were mostly in chains of 10 to 20 joints. 
Generally they were arranged in clusters which often surrounded 
a phagocyte. A few phagocytes containing bacilli were seen. 

No. 41 weighed a little over two kilos. It had three cc. of 
the pepsin solution the day after inoculation, another three cc. 
were injected after a few hours, and again six cc. on the following 
day. It died 71 hours after its anthrax inoculation. The spleen 
was full of bacilli arranged in rows so long as to be difficult to 
count, reminding one of a gelatine culture. Chains of over 30 
segments w r ere noted. 

The above experiments appear to be interesting from several 
points of view, though it must be confessed that they raise more 
questions than they settle. 

In the first place, so far as I know, this is the first time that a 
substance prepared independently of the pathogenic microbe, and 
administered after its advent, has been found to exert an influence 
on the development of the microbe within the body of the animal 
and so on the course of the disease. Since the ferments that I 
experimented with have no special known relation to the anthrax 
bacillus it is to be expected that the same ferments will exert an 
analogous influence on the course of other diseases, which are 
similarly produced by pathogenic micro-organisms. Further I 
have worked with virulent anthrax, a virus which kills 100 per 
cent, of the rabbits inoculated with it. That is to say, the rabbit 
shews practically no power of resisting the onset of the malady. 
May it not be expected that these ferments will shew a still 
greater power of antagonising the microbe in those diseases and 
with those animals in which the mortality is only 10 or 20 per 
cent. ? 

Secondly, the above results are interesting from the point of 
view of the phagocyte theory. The upholders of this theory assert 
that natural or acquired immunity against a disease is due to the 
greater activity of phagocytes. But by the injection of ferments 
I have in some cases at any rate endowed the animal with an 
increased power of resisting anthrax virus. The ferments were 
injected into the blood plasma, and in this liquid the bacilli lived 
and were apparently affected independently of any increased 
phagocyte activity. One of these ferments, the pepsin, is at any 

2—2 



20 Mr E. H. Hankin, On the Injection of Ferments. [Nov. 11, 

rate a post mortem constituent of most of the animal tissues, and 
I think these experiments seem to uphold a theory first sug- 
gested to me by Dr Lauder Brunton, that the "germicidal power" 
that the animal body seems to possess is connected with the 
power it had of producing ferments. This suggestion that I have 
either increased or closely imitated the natural germicidal power 
by ferment injection is supported by the symptons exhibited by 
rabbit No. 37 which finally recovered. As mentioned above, a 
quantity of pus was found at the seat of inoculation. This is 
rendered interesting by the fact that a similar effect is produced by 
inoculating with anthrax an adult rat, an animal which is natu- 
rally refractory to this disease. In the case of rabbit No. 37 and 
of a rat inoculated with anthrax we find a large pus formation at 
the seat of inoculation ; that is to say, not increased activity on 
the part of the leucocytes but an increased degeneration of these 
cells. Does it not seem probable that these cells give out certain 
substances (possibly ferments) which hinder or prevent the growth 
of the bacilli till at length they can be devoured like any other 
inert granules by the active phagocytes ? 

Lastly, these experiments suggest another possibility; namely, 
that by injecting ferments, other microbes which cannot easily be 
cultivated outside the body of the animal may be attenuated 
within it. Possibly in this way attenuated tubercule bacilli could 
be obtained which might be used as a means of vaccinating 
against consumption. 



Note. Since communicating the above results to the Society, 
I have succeeded in obtaining similar results by injection of 
Halliburton's cell globulin after inoculation with anthrax. This 
cell globulin is a proteid, obtained from cells of the lymphatic 
glands, which is either identical with fibrin-ferment or very closely 
connected with it. In my experiments with this substance, the 
elongation of the spleen bacilli was not always so marked as 
with pepsin and trypsin, but longer chains could always be found 
in the lymphatic glands near the seat of inoculation. Perhaps 
more of the individual joints were degenerated than in my former 
experiments, and in cases where the chains did not consist of many 
members, the individual joints were often unusually long. I have 
not yet finished this course of experiments, but the results, as far 
as they go, support the view I have enunciated in my paper con- 
cerning the mechanism of the germicidal power ; and the fact that 
two out of three substances which I have employed are probably 
constituents of leucocytes appears to me to be particularly sug- 
gestive. 

December 7, 1889. 



1889.] Mr W. K. Shaw, On Electrolytes. 21 

November 25, 1889. 
Mr J. W. Clark, President, in the Chair. 

The following were elected Fellows of the Society : 
Arthur Berry, M.A., Fellow of King's College. 
H. F. Baker, B.A, Fellow of St John's College. 
E. W. Brown, B.A., Fellow of Christ's College. 
C. Warburton, B.A., Christ's College. 

The President announced that the adjudicators of the Hopkins 
Prize for the period 1880-82 have awarded the Prize to Mr R T. 
Glazebrook, F.R.S., for his researches in Physical Optics. 

The following Communications were made to the Society : 

(1) On the relation between Viscosity and Conductivity of 
Electrolytes. By W. N. Shaw, M.A., Emmanuel College. 

It has long been suspected that the resistance offered by an 
electrolyte to the passage of electricity through it, depends in 
some way upon the viscosity of the liquid. The mere fact that 
the conduction of electricity is in reality convection by moving 
ions suggests of itself that resistance may be the opposition 
offered by the fluid to the motion of the ions. It is not however 
in any way obvious that the resistance which moving ions would 
meet with would be identical with the ordinary viscosity that has 
to be overcome when a fluid is driven through a capillary tube ; 
so that when G. Wiedemann, in 1856, found that there was an 
analogy between the relative magnitudes of the numbers express- 
ing the conductivity of certain solutions and those expressing the 
fluidity (the reciprocal of the viscosity), the implied relation 
between electric conductivity and fluidity was not at once accepted 
as proved. And indeed, in its crudest form, the hypothesis that 
conductivity is identical with fluidity, as we measure it by Cou- 
lomb's method or Poiseuille's method, or that the two are propor- 
tional, evidently cannot be maintained. For clearly a salt solution 
becoming gradually more and more dilute approaches a finite 
limit of viscosity, namely the viscosity of pure water, whereas the 
conductivity apparently diminishes without limit. Moreover there 
are numerous mobile liquids which do not conduct at all. Nor is 
the mobility which is characteristic of fluids really necessary to 
electrolytic conduction. Professor W. Kohlrausch has examined 
the conductivity of fused salts of silver through a wide range of 
temperature, and he finds that in the case of iodide of silver there 
is no discontinuous change in the conductivity at the melting- 
point, on the contrary the resistance only increases very gradually 
after solidification takes place, remaining less than the minimum 



22 



Mr W. N. Shaw, On the relation between [Nov. 25, 



resistance of H 2 S0 4 , until the mass becomes crystalline and then 
a sudden increase of resistance occurs. The curve representing 
the variation of resistance with temperature for the mixed chloride 
and iodide shews similar properties. Moreover Arrhenius has 
measured the resistance of electrolytes which contain gelatine, and 
these solidify without producing any sudden change of resistance. 
Hence clearly resistance and ordinary viscosity are not the same 
thing. 

Furthermore, in comparing the numerical values of conductivity 
and fluidity we are met with an obvious contradiction of any such 

Conduc- ) JL, 
' tivity 
ofHg 



2 x lO" 1 



io- 2 























I DF 










1 DK 
K. DI 






























F / 










/ 








/ 











C Gramme 
12 3 4 5 ^Equivalents 

( per litre. 

generalisation; for on the addition of an acid or a salt to water 
the viscosity may be increased, whereas the conductivity of the 
solution depends entirely on the presence of the salt or acid. If 
the curves shewing the variation of conductivity with concen- 
tration be compared with those for fluidity and concentration no 
similarity is conspicuous although there are some striking in- 
stances, particularly that of sulphuric acid, of concurrence of pecu- 



1889.] Viscosity and Conductivity of Electrolytes. 23 

liarities which arrest attention. As an example of very normal 
type, I have in fig. 1 reproduced the curve (k) of conductivity 
and concentration in gramme equivalents per litre of NH 4 C1 from 
Wiedemann's Electricitdt, vol. I. p. 610. And to compare with it 
I have plotted a curve, from observations (f) of Grotian*, of the 
fluidity of solutions of NH 4 C1 for the same range of concentration. 
There is no striking resemblance but rather the reverse. In fact, 
the curves for conductivity and concentration are generally roughly 
parabolic in shape with a vertex of maximum conductivity, the 
equations of a series of the curves are given by Wiedemann"]*, and 
the curves for fluidity are according to ReyherJ and Arrhenius§ 
very exactly represented by the equation y = A x , where y is the 
viscosity relative to water, x the number of gramme equivalents 
per litre, and A a constant not differing much from unity. Such 
curves are of course quite different from the conductivity curves, 
and yet, when tables of conductivities and fluidities of different 
solutions are compared, the connexion is very striking; and the 
obvious suggestion is that both fluidity and conductivity are very 
complex phenomena, but that for a given solution they are both 
dependent ultimately in some way upon temperature and con- 
centration, and are so related that, other things being unaltered, 
if the fluidity be increased the conductivity is consequently im- 
proved. It therefore seems more hopeful to compare, as Grotian 
has done, fractional rates of variation of the two quantities 
rather than the quantities themselves. Grotian has chosen the 
rate of variation with temperature, and in parallel columns on pp. 
949 — 952 of vol. II. of Wiedemann's Electricitdt are values of 10 </> 

and 10V where 6 = ( -f- ) — and K = (- Tr ) j~ , for different con- 

W 22 /, 8 \dtJnh 

centrations of solutions of a large number of salts. The suffixes 
indicate temperatures, / is fluidity relative to pure water at 10° C, 
k conductivity referred to mercury, and t temperature. A further 
advantage of this method of comparison is that the units in which 
the conductivity and fluidity are measured do not affect the 
result. 

In order to get a better general view of the comparison of 
the two magnitudes <f> and k, I have examined the curves plotted 
from all the results recorded in Wiedemann, and the general 
parallelism of corresponding curves is very striking. The set 
of lines representing either one of the properties for the solu- 
tions of the different substances traverse the paper at a very 
great variety of inclination to the axes ; some of them are nearly 

* Pogg. Ann., vol. clx., p. 259, 1877. 
t Electricitdt, vol. i., p. 600. 

X Zeitschrift fur Phys. Chem., vol. n., p. 744, 1888. 
lb., vol. i., p. 285, 1887. 



24 Mr W. JV". Shaw, On the relation between [Nov. 25, 

parallel to the axis of abscissae, others are inclined positively 
to it, others again negatively ; but in all cases, with the possible 
exception of acetic acid, which exhibits irregularity, the corre- 
sponding curve for the other property is a curve which though not 
coincident is very nearly parallel, and generally speaking when 
there is a change in the inclination of the curve for one property 
there is a corresponding change of inclination in the curve for 
the other property. When we add to the general appearance of 
parallelism for the lines which present no special peculiarity the 
remarkable parallelism in the exceptional cases of NaHO, the 
curves for which are both very steeply but equally inclined to the 
axis, and of H 2 S0 4 , which furnishes two irregular but still parallel 
curves each shewing a maximum for the same degree of concen- 
tration, the evidence is convincing that there is a real relation 
not of identity, but of parallelism between these temperature co- 
efficients of the two quantities for different concentrations. 

Another means of altering the viscosity without altering the 
other properties of a solution might be found in adding an inert 
non-conducting liquid to the solution and thereby altering the 
solvent. Experiments have been made in this direction by C 
Stephan*, who has measured the viscosity and its temperature 
coefficient for some mixtures of alcohol and water, and the con- 
ductivity and its temperature coefficient for dilute solutions of 
NaCl, KCL LiCl, Nal, KI in these mixtures. The investigation 
differs from Grotian's inasmuch as the temperature coefficients 
of fluidity are not determined for the solutions but only for the 
solvents, so that while the fluidity temperature coefficient seems, 
for the alcoholic solvents, again to be of the same order of 
magnitude and to exceed the conductivity temperature coefficients 
of the solutions, no precise comparison can be made. Stephan's 
comparisons of results are mainly concerned with an enquiry as 
to the constancy of the products of conductivity of the solution 
and viscosity of the solvent for the different solvents ; the con- 
stancy is not established though a limiting value is indicated for 
very dilute solutions. 

E. Wiedemann -f* has moreover compared the conductivities of 
corresponding solutions of NaS0 4 in water and glycerine, but again 
no numerical relation between conductivity and resistance is 
exhibited. 

The relation between resistance and viscosity seems therefore 
not to be a simple one, though the relation between the tempera- 
ture coefficients does seem from Grotian's observations to be 
comparatively simple. With the view of exhibiting this aspect 

* Wied. Ann., vol. xvn., p. 673, 1882. 
t lb., vol. xx., p. 537, 1883. 



1889.] Viscosity and Conductivity of Electrolytes. 



25 



of the question we may consider some of Grotian's observations 
a little more closely. 

I have reproduced in fig. 1 the curves of temperature co- 
efficients for NH 4 C1 plotted from his tables (reducing the abscissae 
to gramme equivalents per litre) and placed them on the diagram 
with the curves for fluidity and conductivity already alluded to. 
The curves for KC1, KBr and KI are strictly analogous, so that 
the diagram may be taken as exhibiting the comparison for a 
group of salts which seem to form a special class of solutions with 
respect to conductivity and fluidity. 

Strict parallelism of the plotted curves would correspond of 

course to a constant difference between the corresponding values of 

1 df 1 dk 

10% -4- and 10 4 -r -=r. These differences are tabulated below, by 
f at k at 

interpolation, for the four salts. 

Table I. 



Concentration, 


NH 4 C1. 


KC1. 


KBr. 


KI. 


per cent. 










5 


34 








10 


29 


30-6 




35-0 


15 


29 


30-0 


28-4 


31-0 


20 


29 


30-4 


30-0 


29-0 


25 




23-8 


28-4 


28-4 


30 






24-0 


28-4 


35 






22-6 


28-0 


40 








27-0 


45 








23-4 


50 








20-0 



The differences are very nearly the same for all the salts for 
concentrations between 10 and 20 p. c, but the two values ap- 
proach one another slightly when the concentration is large. As 
a simple assumption we may take however that the relation 
between the temperature coefficients of fluidity and conductivity 
is one of a constant difference <r, independent of the concentration 
and of the nature of the salt for this particular group of salts. 
It may vary with the temperature and with the nature of the 
solvent, but we will assume for the moment that it is a constant. 
We then get 

1 df_ 1 dk 



fdt kdt 



+ a- 



•(1), 



where a is a small quantity (about 30 x 10 4 ) independent of 



26 Mr W. N. Shaw, On the relation between [Nov. 25, 

concentration but depending on the solvent and possibly also on 
temperature. 

The physical interpretation of this equation would be that 
the effect of temperature upon the conductivity of the solution 
is of a two-fold nature, (1) the conductivity is indirectly increased 
by the increasing fluidity of the liquid and (2) it is diminished 
by some alteration of the properties of the solvent which does 
not affect the fluidity. In electrolytes at ordinary temperatures 
the first effect is predominant, but on very great rise of tempera- 
ture (without secondary alterations of condition) the second effect 
might become very great compared with the first. Thus if the 
solution (with the salt) were volatilized, the conductivity of the 
gaseous mixture might be only a small fraction of the conductivity 
of the solution though the viscosity might have become much 
less. Integrating equation (1) with regard to temperature (as- 
suming <x to be constant) we get 

/o % 

or ' h t =j«f t e-°<t-^ (2). 

Jo 

k 
In equation (2) ~ expresses the relation between conductivity 

J° 

and fluidity at a standard temperature*. If this relation be a 

complicated function of the concentration, as it appears to be, 
there is no reason to infer a general simplicity of relation between 
k and / from the fact of their having temperature coefficients 
which are connected by a simple relation. If we were possessed 
of experimental data that would enable us to refer the properties 
of one class of electrolytes to concentration and temperature as 
variables, in a manner somewhat similar to that in which the 
properties of gases are referred to pressure and volume, further 
insight into the nature of the relation might be obtained. 

The evident relation between viscosity and resistance has not 
yet been satisfactorily accounted for. The hypothesis that the 
motion of the ions, if these be atoms, is opposed by frictional 
resistance which can be measured as viscosity for ordinary motion 
of the liquid seems to be regarded as dubious, although Kohl- 
rausch"f* has shewn " that the supposition of mechanical and electro- 

* Since this paper was read I have seen a paper of Arrhenius (Zeitschr. fur 
Phys. Ghem. Band. iv. Heft. 1, July 1889), in which an equation practically identical 
with (2) is deduced directly from the theory of dissociation. On that hypothesis a 
would be the fractional temperature coefficient of dissociation, and being negative 
would imply, as Arrhenius points out, a temperature of maximum conductivity 
beyond which the temperature coefficient of conductivity would be negative. 

t B. A. Report, 1886, p. 343; Wied. Ann. vol. vi., p. 207, 1879. 



1889.] Viscosity and Conductivity of Electrolytes. 27 

lytic frictional resistance of about equal amount allows the finding 
of an absolute size of molecules which approaches the sizes found 
by other methods." 

But is it necessary to suppose that the individual atoms are 
moved through the liquid ? May the ions not be parts of complex 
molecules of salt and solvent which have to be dragged through 
the liquid? Wiedemann* mentions that such a suggestion has 
already been made but that it is very risky, yet the doctrine of 
electrolysis of molecular aggregates as opposed to that of disso- 
ciated atomic ions has some adherents, and expressions might be 
quoted from the writings of well-known supporters of the dis- 
sociation-theory to shew that they acknowledge the necessity for 
regarding the electrolytic molecule as complex in some special 
cases. I think it may be well to recall that certain phenomena 
may naturally lead to the same view, although I am well aware 
that these phenomena have been otherwise explained in a manner 
that is accepted as satisfactory. 

The view that electrolysis consists in the convection of elec- 
tricity by single atoms only, or their chemical representatives, is 
based upon the resolution of the processes taking place in an 
electrolytic cell into independent phenomena. Electric endos- 
mose, the unequal dilution of solution at the electrodes and the 
deposition of ions are all treated separately. Electric endosmose 
is regarded as the result of the electrification by contact of the 
boundary layer of the solution in the porous partitions which 
divide the cell, and the explanation has been regarded as com- 
plete since von Helmholtz shewed that the difference of potential 
of the boundary layer necessary to explain the effect was not 
more than a few volts. The unequal dilution at the electrodes is 
explained by the theory of migration of the atomic ions with 
unequal velocities, and this explanation has received strong con- 
firmation from Kohlrausch's calculation of resistance, based on 
these atomic ionic velocities and Lodge's experimental verification 
of the calculation in the case of hydrogen. But, apart from 
these reasons, we have no direct evidence that the ions are simply 
atoms, or their chemical representatives. One of the greatest 
desiderata in electrolysis is the determination of the actual ions 
in any case of electrolysis, but it does not seem practicable to 
identify them. Hitherto the tendency has been to assume atomic 
ions if possible, and yet complex ions cannot be excluded. I have 
taken the following cases of electrolysis from Wiedemann f, who 
is himself a strong opponent of the idea of complex molecular 
decomposition in general. 



* See Electricitat, vol. it., pp. 953, 962. 
t lb., vol. ii. 



28 



Mr W. N. Shaw, On the relation between [Nov. 25, 



We see therefore that in spite of the present tendency to 
reduce the electrolytic action to convection by atomic ions if 
possible, there are many cases in which the ions are aggregations 
of atoms, if not strictly molecular, and in some cases molecules 
are associated with a moving atom in electrolysis. It seems 
therefore not altogether unreasonable to assume that the decom- 
position of a complex molecule may not be exceptional but the 
general rule. Let us therefore take the venturesome step of 
considering that the whole result of electrolysis consists in the 
separation of complex molecular aggregates of salt and water each 
into two parts, each part containing one dissociated atom or its 

Table II. 



+ Ion. 


Electrolyte. 


-Ion. 


K 


KHO 


HO 


H, NH 3 


NH 4 C1 


CI 


K 2 


K 2 Cr 2 7 


2Cr0 3 , 


Na 2 


Na 2 HP0 4 


|(H 2 o, p 2 o 5 , eg 


uo 2 


uo 2 ci 2 


ci 2 


K 4 


K 4 Fe(CN) 6 


Fe(CN) 2 ,(CN) 4 


Ag 


Ag 2 CN 2 


AgCN, ON 


K 


an j 

K J^ 


Cdl 2 , I 


Cd 


2(CdI 2 ) 
(in alcohol) 


Cdl, I 


Cd 


3CdI 2 

(in alcohol) 


2CdI, I 



chemical representative, and suppose that the complete result of 
electrolysis, including the transference of liquid known as endos- 
mose, the migrations of ions, and the deposition on the electrodes 
may be accounted for by this splitting up of the complex molecule 
into two parts and the transference of the separated parts in 
opposite directions, so that each separated atom would be loaded 
with a number of water molecules or salt molecules or both. If 
this be assumed, the effect of a porous diaphragm in a solution 
would be not to cause the transference of liquid but to prevent 
it slipping back again, and the friction against the plug which 
would prevent the slip back would be very similar in its mathe- 
matical expression to the force supposed by von Helmholtz to 
cause the transference. 

The reasoning by which Wiedemann, on p. 592 of vol. II. of his 
Electricitat, shews that electric endosmose is independent of 
migration seems to me to be as follows. The total gain of cation 



1889.] Viscosity and Conductivity of Electrolytes. 



29 



in the cathode vessels is not the same for different degrees of 
dilution when there is a porous diaphragm, but if you subtract 
the amount due to the increase of volume of solution, the amounts 
are approximately the same; hence if the increase of volume be 
regarded as an entirely independent phenomenon the cations may 
be assumed to be the same ; viz. atoms of copper, for all the dif- 
ferent degrees of dilution. But the same result would be arrived 
at if we assumed that in the more dilute solutions a greater 
number of molecules were associated with the atoms. Assuming 
that of the salt which is decomposed half is taken from the 
anode vessel and half from the cathode vessel, I have calculated 
the molecules that must be decomposed to give the required total 
gain at the cathode which is tabulated for CuS0 4 by Wiedemann 
for solutions of different strengths. They are as follows 

Table III. 



Concentration 

in grammes 

of Copper 

per 100 cc. 


Molecule decomposed. 


3-3793 

3-118 

2-263 


Cu„(CuS0 4 , 5H o 0)/(S0 4 ) o 
Cu 2 (CuS0 4 , 6H" o 0)/(S0 4 )l 
Cu 2 (CuS0 4 ,8H 2 0)/(S0 4 ); 



The molecules which are actually decomposed may be more 
complex than those given, by combination with molecules of solu- 
tion, but the association of such molecules with the ions would not 
affect the ultimate relative distribution of the electrolyte. More- 
over all the ions in a specific solution need not be of the same 
order of complexity. 

It would be a natural consequence of this view to suppose 
that when the solutions became very dilute, the number of mole- 
cules of water associated with the moving atoms would be very 
large, ultimately being proportional to dilution ; in that case the 
electrolysis would depend mainly on the motion of these large 
molecular aggregates past each other and the resistance would be 
of the nature of ordinary viscosity. Under these circumstances 
the number of molecules associated might depend on the number 
and be independent of the nature of the atoms and the ionic 
velocities, and the resistance of all electrolytes would tend to the 
same value as they are known to do in solutions of extreme 
dilution*. 



* F. Kolrausch, Gegemoartige Anschauung, <£-c., p. 28. 



30 Mr W. N. Shaw, On Electrolytes. [Nov. 25, 

I have not hitherto dealt with the difficulty raised by the 
remarkably accurate calculations of resistance of electrolytes from 
ionic velocities based on the assumption of atomic ions and verified 
experimentally by Lodge in the case of hydrogen. In the first 
place however I find it difficult to realise the conception of an ex- 
tremely large number of small bodies (atoms) moving in opposite 
directions through an aggregation of other small bodies (the solu- 
tion). There is also an objection to Kohlrausch's theory on the 
ground that it assumes all the atoms of the dissolved salt to be 
moving. It seems to me that the conception is easier if we regard 
the whole of the solution as divided between the dissociated atoms ; 
the effect of the electromotive force would then be to pull the whole 
of one set of atoms with their associated molecules in one direction 
and the whole of the other set with their associated molecules in 
the opposite direction, and we thus get a stress shearing the one 
set of molecules past the other set ; the result will be a relative 
motion of the atoms carrying their loads and the mean velocity 
will depend on the electromotive force and the viscosity ; but 
the motion is relative; as to the absolute velocity of each set 
the velocities in the two directions may be regarded as equal. A 
particular atom may at one time be a dissociated one moving to 
meet a partner with this velocity, at another time it may belong to 
an associated molecule and be travelling with another dissociated 
atom in the same direction as before or in the opposite ; to deter- 
mine the mean velocity of all the cation atoms for instance, in 
one cross-section, we must deduct the number of backward steps 
it takes in the unit of time as part of a molecule associated with 
an anion atom from the number of forward ones it takes either 
as a dissociated atom or part of a molecule associated with a 
cation; this mean velocity will be equivalent to a transference of 
the atom through the solution. It is this mean rate of trans- 
ference which must be always the same for the same atom in 
very dilute solutions, no matter with what other atom it was 
associated as a salt, and it is this mean velocity which Lodge has 
measured. 

I think, therefore, that it may be possible to frame a general 
theory of electrolytic action on the basis of a hypothesis of com- 
plex molecular aggregates, dissociated in a solution, or separated 
by the current, into ions consisting of atoms with attached mole- 
cules ; and such a theory might explain all the various electrolytic 
phenomena, including migration, endosmose, and the relation 
between viscosity and resistance. I am well aware that the brief 
sketch contained in the foregoing paper cannot be regarded in 
any way as a complete statement of such a theory, and that 
further application in detail is necessary before the theory can 
claim to be satisfactory. All that I venture to say is that there 



1889.] Mr C Chree, On Elastic Solids. 31 

are no crucial experiments, that I am acquainted with, -which 
definitely and finally contradict it, and until such experiments are 
made it may be well to bear in mind the possibility of such 
explanations as the theory affords. 

(2) The finite deformation of a thin elastic plate. By A. E. H. 
Love, M.A., St John's College. 

(3) A solution of the equations for the equilibrium of elastic 
solids having an axis of material symmetry, and its application to 
rotating spheroids. By C. Chree, M.A., King's College. 

[Abstract] 

A solution is obtained of that type of the elastic solid equations 
which contains five elastic constants, answering to those bodies in 
which the structure is symmetrical round an axis. The solution 
proceeds in ascending powers of the variables x, y, z. In the 
expressions for the displacements terms containing powers of the 
variables below the fourth are retained. Thus the solution, while 
complete so far as it goes, can solve exactly only certain classes of 
problems. One of the problems which it can completely solve is 
that of a spheroid of any eccentricity rotating uniformly about 
the axis of revolution, this axis being in the direction of the axis • 
of symmetry of the material; and it is to this problem that atten- 
tion is mainly directed. 

The solution obtained for the general case of a rotating sphe- 
roid being somewhat complicated, certain special cases are first 
considered. The first of these cases, that of a very flat oblate 
spheroid, applies approximately to a thin plate rotating about 
the normal to its plane through the centre. The second case, that 
of a very elongated prolate spheroid, applies even more satis- 
factorily to the non-terminal portions of a rotating cylinder of 
length great compared to its diameter. In these two cases the 
material is of the 5-constant type. The third special case is that 
of uniconstaut isotropy in spheroids of every form. 

From the light thrown on the question by the results obtained 
in the third special case it becomes possible to dissolve out of the 
complicated mathematical expressions for the general case a very 
considerable amount of information as to the state both of stress 
and strain throughout the spheroid. The key to this information 
is supplied by the recognition in every material whether of the 
5-constant or of the isotropic type of a "critical" spheroid. The 
ratio of the "polar" to the "equatorial" diameter of this spheroid 
depends only on the elastic constants of the material, and is given 
by a simple expression. The following, which are only a few of 



32 Mr H. F. Baker, On the Concomitants [Nov. 25, 

the results obtained, will show the importance of the critical 
spheroid: — 

In any 5-constant or isotropic rotating spheroid one of the 
principal stresses is everywhere perpendicular to the " meridian " 
plane, and of the two in the meridian plane the greater makes an 
obtuse or an acute angle with the perpendicular on the " polar " 
axis produced outwards according as the spheroid is more or less 
oblate than the critical spheroid. In particular the surface is under 
a tangential tension in the meridian plane in the former case, but 
under a compression in the latter. In the critical spheroid one of 
the principal stresses is everywhere zero, and on the surface there 
is no stress at all in the meridian plane. In any species of bi- 
constant isotropic material, for a given value of the equatorial 
diameter, the critical spheroid is the form in which the " tendency 
to rupture " on Saint- Venant's theory is the greatest. 

In the case of uniconstant isotropy the character of the strain 
throughout rotating spheroids of all shapes is completely investi- 
gated, and is shewn in a table. In the general case of 5-constant 
material a similar, though not so exhaustive, analysis is given. 
Tables supply the values of the changes in the lengths of the 
equatorial and polar diameters, and the strains at the centres for 
various kinds of biconstant isotropic materials in spheroids of 
various forms. 

The variations of some of the more important quantities are 
also shewn graphically. 

(4) On the concomitants of three ternary quadrics. By H. F. 
Baker, B.A., St John's College. 

[Abstract] 
The author applies a modification of the symbolical method 
suggested by Clebsch and Gordan (Math. Annul. I. 90 and I. 359) 
to obtain the set of concomitants in terms of which all the system 
are expressible as rational integral algebraic functions. The 
result is given by the following table. The forms are taken 
respectively to be 

a 2 - a ' 2 = a " 2 = 

U x U x U x '"> 

r 2 — c ' 2 = c " 2 = 

V x — V x — L x — 

Also (aa'uf is abbreviated into iC = u/ = u a - z =...', 
and (aa'a") 2 into a a etc.: 

and so for the other two forms. 

Such a symbol as (523) preceding a form indicates that the 
form is of the fifth degree, second class, and third order. 



1889.] 



of three Ternary Quadrics. 



33 



Only one form of a given type is written down ; the others 
may be obtained by interchanging the letters — the number of 
forms so obtainable is given by the number in brackets which 
follows. The forms are arranged in sets, as obtained, according to 
their degrees. 



0. (Oil) 


= u x 


(1) 


3. (300),= 


= a 2 


(3) 


1. (102) = 


= a x * 


(3) 


(300),= 
(300) 3 = 


= b 2 
= (abc) 2 


(6) 
(1) 


2. (212) = 


= (bcu) b x c x 


(3) 


(311),= 


= u a b a b x 


(6) 


(220), 


= u 2 


(3) 


(811), 


= (abc) (bcu) a x 


(3) 


(220) 2 


= (bcu) 2 


(3) 


[303] = 


- (abc) a x b x c x 


(1) 








(330) 


= (bcu) (cau) (abu) 


(1) 


4. (410) = 


= (bcu) b a c a 


(3) 








(402) f = 


= b a c a b x c x 


(3) 


5. [501], 


= (abc) a x b a c a 


(3) 


(402), 


= (Pyx) 2 


(3) 


(501),= 


= (Pyx) apa y 


(3) 


(421), 


= (bcu) b a c x u a 


(6) 


(520) = 


= UpUyapay 


(3) 


(421), 


= (bcu) b y c x u y 


(6) 


(512),= 


= (Pyx) apa x Uy 


(6) 


[421] 3 


= (a'bc) (uca) (uab) a a 


'(3) 


[512], 


= (abc)apUpb x c x 


(6) 


(421) 4 


= (Pyx) U0y 


(3) 


(612), 


= (Pyx) c x cpu y 


(6) 


6. (600) 


= («PyT 


(1) 


7. [710],: 


= (aj3y)apa y u a 


(3) 


(61D, 


= (a/37) (#7*0 u * 


(3) 


[710], 


= (bcu) apa y b y Cp 


(3) 


[6H] 2 


= apOybybsUp 


(6) 


[721] 


= ( a Py) b„b x upUy 


(6) 


[630], 


= (a/37) UaUpU y 


(1) 








[630], 


== (abu)apbyiipu y 


(6) 


8. [801],= 


= (Pyx) b y cpb a c a 


(3) 


[630] 3 


= (bcu) UpiiybyCp 


(3) 


[801], 


- (a'bc) apttybyCpaJ 


(3) 


(603), 


= (Pyx) (7a*) (apx) 


(1) 


[812] = 


= (a'fiy) (yax) (aj3x)u 


*(8) 


[603], 


= (Pyx) a x b x apby 


(6) 


9. [911] = 


= apayb y b a c a c x up 


(6) 


[603] 3 


= {pyx) b x c x b y cp 


(3) 









[1010] = (a'Py) b y Cpb a c a u a , (3) 
The degree — class — order symbols of eighteen of the types are 
placed in square brackets. This indicates that they are reducible 
after multiplication by u x . Some of them are further reducible 
on multiplication by u 2 : namely these are (501),; (710),; (801),; 
(911) ; (1010). Allowing these reductions the system is expressible 
by 13 kinds of forms, viz. by 

((abc) 2 ba a 2 a 2 (bcu)b x c x (bcu) 2 u a b a b x (abc) (bcu) a x 

{(a/37) 2 W (fiyx)upUy (pyx) 2 (a/3y)(Pyx)u a 

(bcu) (cau) (abu) (bcu)b a c a b a c a b x c x (bcu)b a c x u a (bcu)b y c x u y 

(fiyx) (afix) (yax) (Pyx)a p a y UpU y apa y (Pyx)apa x u y (Pyx)c x CpUp 

VOL. VII. PT. I. 3 



34 Mr H. F. Baker, On Ternary Quadrics. [Nov. 25, 1889. 

where a form and its leciprocal, as well as forms of the same 
type, are counted as being of the same kind. 

Note. There are often identical integral relations among 
forms of the same type. These are not here set down. Further, 
if we allow algebraic functions of any rational kind, all the simul- 
taneous concomitants can be expressed in terms of fifteen con- 
comitants (Forsyth, American Journal of Mathematics, XII. p. 54). 



COUNCIL FOR 1889—90. 

President. 
John Willis Clark, M.A., F.S.A., Trinity College. 

Vice-Presidents. 

E. J. Routh, Sc.D., F.R.S., Peterhouse. 

C. C. Babington, M.A., F.R.S., Professor of Botany. 

G. D. Liveing, M.A., F.R.S,, Professor of Chemistry. 

Treasurer. 
R. T. Glazebrook, M.A., F.R.S., Trinity College. 

Secretaries. 

J. Larmor, M.A., St John's College. 
S. F. Harmer, M.A., King's College. 
A. R. Forsyth, M.A., F.R.S., Trinity College. 

Ordinary Members of the Council. 

J. E. Marr, M.A., St John's College. 

J. C. Adams, Sc.D., F.R.S., Lowndean Professor. 

A. S. Lea, Sc.D., Gonville and Caius College. 

T. McK. Hughes, M.A., F.R.S., Woodwardian Professor. 

J. W. L. Glaisher, Sc.D., F.R.S., Trinity College. 

W. N. Shaw, M.A., Emmanuel College. 

W. Gardiner, M.A., Clare College. 

W. Bateson, M.A., St John's College. 

A. Cayley, Sc.D., F.R.S., Sadlerian Professor. 

G. H. Darwin, M.A., F.R.S., Plumian Professor. 

W. J. Lewis, M.A., Professor of Mineralogy. 

W. H. Gaskell, M.D., F.R.S., Trinity Hall. 



PROCEEDINGS 



OF THE 



€Kmhx&Qt |j{)il0sap{jkal Sbrkijr, 



January 27, 1890. 
Mr J. W. Clark, President, in the Chair. 
The following were elected Fellows of the Society : 

J. G. Adami, M.A., M.B., Christ's College. 
T. Roberts, M.A., St John's College. 
K. H. Solly, M.A., Downing College. 
E. H. Hankin, B.A., St John's College. 

The President called attention to the proposed International 
Memorial to Dr Joule, late Honorary Fellow of the Society, and 
announced that subscriptions might be sent to the Honorary 
Secretary, Joule Memorial, Royal Society of London, or that they 
would be received by Mr R. T. Glazebrook, Treasurer of the Cam- 
bridge Philosophical Society. 

The following Communications were made to the Society : 

(1) Non- Euclidian Geometry. By Prof. Cayley. 

{Abstract.) 

The chief object of the Memoir is the development of the 
analytical theory : and as the form assumed for the equation of 
the Absolute is x 2 + y 2 + z 2 + w 2 = 0, the formula? obtained may be 
regarded as belonging to Elliptic Space. But this is not the point 
of view of the Memoir ; the space considered is ordinary space, it 
is only the notion of distance (linear, angular, and dihedral) which 
is modified. Thus for instance, lines perpendicular to each other, 

VOL. VII. PT. II. 4 



36 Mr Larmor, On Rigidly connected Points, [Jan. 27, 

in the ordinary sense, exist, but there is no occasion to consider 
them : in place thereof we consider lines which are in the new 
sense perpendicular to each other, and the theory is an entirely 
distinct one ; given any two lines, we have perpendicular to 
each of them (not a single line, but) two lines, or say there are 
two perpendicular distances : the theory of these distances is con- 
sidered in some detail. 

(2) A Scheme of the Simidtaneous Motions of a system of 
Rigidly connected Points, and the Curvatures of their Trajectories. 
By J. Larmor, M.A., St John's College. 

The following analysis is suggested by the theorems of 
De la Hire and Savary, whereby the determination of the cur- 
vatures of the trajectories of the different points of a solid 
moving in one plane is reduced to geometrical construction. 
In this theory the construction is based on the circle which at 
the instant in question is the locus of points for "which the 
curvature is zero, the well-known circle of inflexions. See 
Williamson's Differential Calculus, Chapter xix * 

In the generalized theory, when the motion of the solid 
is not confined to be uniplanar, the first problem is to determine 
the nature of the locus of inflexions. This is easily effected by 
kinematical considerations ; for the criterion of a point x, y, z 
beinsr on the locus is that its acceleration is in the same direction 
as its velocity, viz. that 

- = ^ = -\. ...(1). 

Now we may specify the motion of the solid by u, v, w the 
components of the velocity of the origin, and w x , co y , co z the 
component angular velocities of the body round the axes of 
coordinates. Then, as usual, 

x = u-yco !l + ZG) v (2), 

x = u — ycb z + zo) y — m z (v — zw x + xco x ) 

+ co y (w-xo) y + ya) x ) (3), 

with two pairs of other similar formulae. 

The equations of the curve of inflexions are now obtained by 
substitution in (1). 

* I find that questions similar to the ones here discussed are analyzed by the 
method of vectors from a fixed origin in the Comptes Rendus, 1888, pp. 162 — 5, by 
Gilbert, who also gives references to other writers on this subject. His investiga- 
tions relate chiefly to the case when a point of the system is fixed. 

The principal results obtained in this note have been stated in the Cambridge 
Mathematical Tripos, Part II., June 1, 1889. (Camb. Exam. Papers, 1888-9, 
p. 569.) 



1890.] and the Curvatures of their Trajectories. 37 

The result will be simplified if we take the central axis of 
the motion for the axis of x, so that u = V, co x = Q, while the 
other components vanish, though their fluxions remain finite. 
We thus obtain the equations 

L M-n 2 y N-il'z 

- = = (4), 

-37 z — y y 

wherein 

L = u — yw z + zw>^\ 

M = v — zco x + xw\ (5), 

N = w - xw v + y&J 

and «r is written for V/fl, the pitch of the given screw-motion. 

The equations (4) thus obtained may readily be verified by 
intuition ; for L, M, N represent the component accelerations 
due to the motion of the origin and the change of values of 
the angular velocities, while 0, — 12 2 ?/, — Q?z are the components 
of the centrifugal force round the central axis, and it is clear 
that these together make up the total acceleration. 

These equations (4) represent the curve of intersection of 
two paraboloids. To reduce them to the simplest possible form, 
first turn the axes of y, z round that of x, so as to make w. zero. 
Then move the origin along the central axis a distance h, so 
that the equation referred to this new origin is obtained by 
writing x + h in place of x, and take h = wjm y . We thus have 
finally 

ii.+ zw y _v — Z(b x — Q?y — xcb,^ yco^— fl^x . 

\")i 

■sr z —y 

where we notice by the way that the numerators give the simplest 
form to which the rectangular component accelerations for a 
moving solid can be reduced. 

The equations (6) represent the curve of intersection of a 
parabolic cylinder having its generators parallel to the axis of x 
with a rectangular-hyperbolic paraboloid having its axis in the 
same direction. These surfaces intersect on the plane infinity 
along the line where any plane z — constant meets it. The finite 
part of their curve of intersection, which is the proper inflexional 
curve, is therefore a tivisted cubic. Its equations (6) may be put 
in the form 

y=A + Bz(a + /3z)\ 
■x = y(a + (3z) J ^ h 

which are unicursal in the parameter z. 

We remark that a wire of this form is the most general solid 
that can be moved with rotation so that all its points are in- 
stantaneously describing straight paths ; also that any wire whose 

4—2 



38 Mr Larmor, On Rigidly connected Points, [Jan. 27, 

form is given by (7) possesses this property, the movement being 
a screw of pitch — /3" 1 (1 + B~ l ) round the axis of x, eased off in 
a way that retains one degree of indeterminateness. 

We proceed to investigate the trajectory of any point of the 
solid by the aid of this cubic. 

Through any point, as is well known, one and only one chord 
of the cubic can be drawn. We may regard this chord as a line 
of constant length moving with its extremities on two fixed 
lines, which may be considered straight so far as the determi- 
nation of accelerations and curvatures is concerned. 

Consider two consecutive positions of it, BG and B'C; let 
BP = B'F = p, and CP = C'P' = p, and let p + p' = a. Complete 




Fig. 1. 

the parallelogram C'CBA, and draw P'M, P'N parallel to A B', 
AC, as in Fig. 1. The circumstances of the motion are given 
by the velocities of the extremities of BG ; let then 

BB' = bt+±bt 2 \ 8) 

cG' = ct + yf\ { 

so that b, c are the velocities, and b, c the accelerations of B and G 
along their straight trajectories. 

The point Q moves in a fixed plane which is parallel to both 
BB' and GG', being parallel to the plane ABB'. The coordinates 
of Q referred to axes of x and y parallel to BB' and GG' are the 
same as the coordinates of N referred to axes BB' and BA . They 
are therefore given by 

BB'=-x=bt + ibf] 

9 (9). 

CG'= p -y=ct + ±c? I 



1890.J and the Curvatures of their Trajectories. 39 

These are the equations of the path of P, correct as far as the 
second order, and referred to the parameter t. 

The determination of the radius of curvature E at the origin 
may now be made by the usual methods, bearing in mind that 
the angle between the axes is &>, the angle between BB' and GO'. 
It is sufficient to give the result 

R ( by 2 + cy+2bcppcos^ 

a sin a (be — be) pp 

for its interpretation suggests a purely geometrical method of 
arriving at it, as follows. 

The velocity V of P' is the same as that of N, and is therefore 

the resultant of velocities — b and - c parallel to BB' and BA : 

a a L ' 

thus 

V 2 = - 2 (b 2 p 2 + c 2 p 2 + 2bcpp' coso) (11). 

CI 

In the same way, the acceleration / of P' is the resultant of 
accelerations — b and - c parallel to BB' and BA ; its value may 

Qj Co 

therefore be written down. 

Also, if 6 denote the angle between V and f, we have Vf sin 9 
equal to the area of the parallelogram contained by the vectors 
representing Fand/; therefore 

Vfwa.6 = (£ b?-6-£c P -b)sma> 

\a a a a J 

= £§ (bc-bc)s'mo3 (12). 

Now by Huygens' fundamental formula of centripetal acce- 
leration in a curve, 

/sm0=^- (13), 

therefore we arrive at the formula (10), which may also be 
written 

a 2 V 3 

R= ~ . —, (14). 

(be - be) sin co PP 

We may exhibit the result in a geometrical form. 




40 Mr Larmor, On Rigidly connected Points, [Jan. 27, 

Draw 0/3, Oy to represent the velocities of 
B and in magnitude and direction; divide 
the fixed line /3y in v so that ftv is to vy as 
BP to CP. Then Ov represents the velocity 
of P in magnitude and direction, and 

R = k^- (15), 

pv . yv 

where k represents the constant factor 

(be — cb) sin w 

It is worthy of notice that altering the accelerations of B or G 
only alters the value of the factor k ; so that the curvatures of 
the trajectories of all points on BO are altered in the same ratio. 

We have thus from Fig. 2 a complete specification of the 
velocity and curvature for any point P on BC; for the plane of 
the trajectory is that parallel to the directions of motion of B 
and 0. The acceleration of P may be constructed from those of 
B and C in the same way as the velocity. The value of the 
curvature involves the accelerations of B and C only as entering- 
into k. A known value of the curvature at any one point de- 
termines k once for all, and the values of these accelerations are 
no longer necessary. A geometrical form for k is given by (21). 

The construction of Fig. 2 applies to any line in the moving 
solids in so far as the determination of velocities only is con- 
cerned ; for the determination of accelerations and curvatures it 
applies only to a chord of the curve of inflexions, as above. 

If however the planes of the curvatures at any two points 
on the line are parallel, it must be such a chord. This may 
be established by an easy extension of the method of Fig. 1. 
For taking B, to represent these two points, we have now 
BB' and also CO' and BA curved instead of straight lines, and 
we obtain a quadratic equation giving two positions of N on 
AB' for which the curvature of the path of P is zero ; this gives 
two points, real or imaginary, on BO, which are also on the curve 
of inflexions. 

The formula (10) for the curvature should be reducible to 
a form depending only on the geometry of the diagram. In 

fact, if BB' = h, CC' = k, B'BC = /3, C'OB = y, 

we have 

B'C' 2 =a? + h 2 + k* — 2ah cos /3 - 2ak cos y — 2hk cos m, 
so that, as B'C = a, h is determined in terms of k by the equation 
h 2 — 2h (a cos /3 + k cos &>) — 2ak cos 7 + k 2 = (16), 



1890.] and the Curvatures of their Trajectories. 41 

which gives, to the first order, 



7 7 cos 7 ., K . 

h = -h —± (17), 

cos/y v ' 



to the second order, 



/„ k cos co\ f n , 7Q TO cos 2 7 



h = 9 ( 1 -= ) ( 2ak cos 7 — k* — Tc 



2a cos (3 V a cos /3/ \ cos 2 /8 

— ~h ~ + ~— - , ~ (cos 2 /3 + cos 2 7 + 2 cos /3cos 7 cos co). . .(18). 

cos p 2a cos p 

Now by (8) we have to the same order 

c 2 V c/ 

-'*+^* (19)- 

Comparing these, we have 

b cos /3 = — ccos 7 (20), 

c 3 
6c — be = 3-5(cos 2 /3 + cos' 2 7+ 2 cos /3 cos 7 cos co)...(21). 

Substituting in (10), 

p _ (|0 2 cos 2 /5 + p' 2 cos 2 7 — 2pp' cos /3 cos 7 cos co)- 99 . 
pp sm co (cos p + cos 7+2 cos p cos 7 cos co) 

where /3 and 7 are the angles which BO makes with. the curve 
of inflexions at B and C, and co is the angle between the tangents 
to the curve at those points ; so that the curvature is expressed 
in terms of purely geometrical quantities. 

It is easy to verify that this theory leads to the correct results 
for uniplanar motion. For in (4) we have now w = 0, co = 0, co\— ; 
the curve of inflexions is therefore a circle passing through the 
central axis /. If now the special chord PI meet this circle 
again in Q, we may apply (14) if we write in it 6=0, b = to 2 IC; 
thus we obtain (and still more directly from (22)) the correct 
result 

R.PQ = IP 2 (23). 

The theory for uniplanar motion may also be reduced to 
simple kinematic considerations as follows. There is one point 
connected with the solid which has no acceleration ; by taking 
this point for origin and reducing it to rest in the usual manner, 
we see that the acceleration of any other point with respect to it, 



42 



Mr Bateson, On the perceptions 



[Feb. 10, 



i.e. in this case the total acceleration, is made up of a com- 
ponent ra transverse to the radius vector and a component rco 2 
Therefore the resultant acceleration is 



towards the origin 

o 



(a) 4 + 6?f r, 



and it acts at a constant inclination a to the r; 
is given by tana=&)/&) 2 ; a known theorem. 
Thus if / denote the instantaneous centre of 
the motion, /' this centre of accelerations, and 
P any point, so that 

IP = r, i'P=p, lpl' = 0, 

we have, by the theorem of centripetal ac- 
celeration, 



lius vector, which 




Kg. 3. 



therefore 



(co 4 + 6> 2 f p sin {6 - a) = 
r 2 sin a 



R 



R = 



p sin (6 — a) ' 

The points whose paths have zero curvature are given by 6 = a, 
and therefore lie on a circle through / and /', the circle of in- 
flexions. 

Let this circle cut PI in Q ; then 

p sin (0 — a) = PQ sin a, 

therefore E = IP 2 /PQ. 



February 10, 1890. 
Professor Babington, Vice-President, in the Chair. 

The following Communications were made to the Society : 

(1) On the perceptions and modes of feeding of fishes. By 
W. Bateson, M.A., St John's College. 

In the course of observations made at Plymouth and elsewhere 
it appeared that the majority of Fishes are diurnal in their habits 
and seek their food by sight, but that a minority are almost 
entirely nocturnal and hunt by scent. To the latter class belong 
Protopterus, Skates and Rays, the Rough Dogfish, Sterlet, Eel, 
Conger, Rocklings, Loaches, Soles, &c. These creatures remain 
buried or hidden by day but career about at night in search of 
food, returning to their own places at dawn. If while they are 



1890.] and modes of feeding of Fishes. 43 

thus lying hid, food or even the juice of food-substances is put 
into the water, they come out after an interval and search vaguely, 
without regard to the direction whence the scent proceeds. Some 
of the animals (Rocklings, Sterlet) have special tactile organs in 
the shape of barbels or filamentous fins with which they investi- 
gate their neighbourhood, while others (Conger and Eels) feel 
about with their noses. None of the fishes which hunt by scent 
seem able to recognise food by the sense of sight, even though it 
be hanging freely before their eyes. 

The mode of feeding of the Sole is peculiar. When searching 
for food its skin is more or less covered with sand, which renders 
it inconspicuous when moving on the bottom. This sand adheres 
to mucus which is probably exuded when the smell of food is 
perceived. The Sole seeks its food exclusively on the bottom, 
creeping about and feeling for it with the lower side of its face. 
If a worm is lowered by a thread until it actually touches the 
upper side of the head of a Sole, the animal is still unable to find 
it but continues to feel for it on the sand. There is however no 
reason to suppose that the sight of these fishes is deficient. A 
Rockling at Plymouth had already learnt to come out to be fed if 
any one came near the tank, though it still did not recognise a 
worm swimming in the water. Particulars were given of the 
various irideal mechanisms which occur among fishes. 

This investigation was undertaken at the instance of the Marine 
Biological Association as a preliminary step towards improving the 
supply of bait. The experience gained suggests that a bait for 
the south coast, where Conger and Skate are chiefly caught, could 
be made by extracting the flavour of Squid or Pilchard and com- 
pounding it with a suitable ground-substance. Though few practical 
experiments were made, it was found that an ethereal extract of 
Nereis or Herring, for example, greatly attracted some of these 
fishes. 

(2) Notes on Lomatophloios macrolepidotus (Goldg.). By A. C. 
Seward, M.A., St John's College. 

[Received January 25, 1890.] 

In the Zeitschrift der deutschen geologischen Gesellschaft (Band 
xxxiii. p. 354) Prof. Weiss of Berlin gives a short account of an 
interesting fossil plant from Langendreer in the Westphalian 
coalfield: it is preserved in Siderite ("Spatheisenstein") and, ac- 
cording to Weiss, is a cone-like specimen having the characteristic 
external features of Lomatophloios macrolepidotus (Goldg.); size 
of specimen 18 cm. long and 13 cm. broad, with a fairly uniform 
thickness of about 3'5 cm. 1 

1 The form of the specimen and the character of the leaf-bases are shewn in 
figs. 3 and 4 (Plate III.). 



44 Mr A. G. Seward, On Lomatophloios macrolepidotus. [Feb . 10, 

From an examination of microscopical sections Weiss came to 
the conclusion that the structure was that of a fruit-cone of 
peculiar organisation. The internal structure is thus described 1 : 
" From an axis of considerable breadth (about 12 cm.) proceed 
the lower parts of the leaf organs which have the well-known 
rhombic leaf-bases with transverse leaf-scars at their upper end. 
The leaf-bases are sack-like and arched towards the lower end, 
from which they curve upwards and outwards : in their lower 
part they are much inflated, and almost reach to the leaf-base 
next above ; they then gradually become narrower until they are 
of the same breadth as the leaf-scars ; this inflated lower portion 
encloses a sack- or flask-shaped space. The organisation can be 
easily made out as the tissue is in part well preserved. The 
flask-shaped spaces contain large (sometimes 2 '5 mm. in diameter) 
round and elliptical bodies, sections of which shew them to be 
bounded on the outside by a wall formed of polygonal cells and 
enclosing in their interior numerous grains." These round and 
elliptical bodies are considered by Weiss to be sporangia full of 
spores. After this description Weiss goes on to say that the 
specimen must be regarded as a fruit-cone of Lomatophloios with 
an internal structure comparable to that of Isoetes. The same 
specimen is also briefly described in the Botan. Gentralblatt 2 . In 
his small book on the coal-measure flora 3 Weiss gives a figure of 
Lomatophloios macrolepidotus, which is described in the text as 
a large cone under the name of Lepidostrobus macrolepidotus. 
Solms-Laubach 4 refers to the description of the same fossil, as 
given by Weiss, and, judging from the extraordinary size of the 
axis, throws out the suggestion that in this case we probably have 
a fructification which was borne on the leaves of the main stem, 
and not on a fruit-bearing branch. Schenk 5 mentions Lepidophloios 
macrolepidotus (Weiss) as an example of a cone with a very stout 
axis, and probably belonging to Lepidophloios. Prof. Weiss has 
unfortunately never published any figures to illustrate his descrip- 
tion of the internal structure of this so-called Lomatophloios or 
Lepidodendron fructification. When recently going through the 
collection of fossil plants in the Berlin Bergakademie I saw the 
specimen and also had an opportunity of examining micro- 
scopically some of the prepared sections 6 . 

The microscopic characters are briefly noticed by Weiss : one 
or two additional points may however be mentioned. On each of 
the leaf-bases there is a small indentation (Plate III. fig. 4, i) 

1 Zeitschrift der deutschen geol. Gesell., Bel. xxxiii. p. 355. 

2 Botan. Gentralblatt., Vol. viii. p. 157. 

3 Flora der Steinkohlenformation, Fig. 33, and p. 7. 

4 Einleitung in die Paldophytologie, p. 241. 

5 Die fossilen Pflanzenreste, p. 70. 

6 I had access to the specimens through the kindness of Prof. Weiss. 



1890.] Mr A. G. Seward, On Lomatophloios macrolepidotus. 45 

immediately below the leaf-scars (fig. 4, s) ; on the leaf-scars 
I did not notice any definite traces of vascular bundle-scars such 
as are represented in Weiss' figure 1 (this figure is not taken from 
the specimen in the Bergakademie, but seems to be copied from 
the one given by Goldenberg) 2 ; the absence of traces of these 
bunclle-scars is probably accidental, and due to imperfect preserva- 
tion. The lower parts of the leaf-cushions have a more or less 
wrinkled appearance, in fact they remind one strongly of the per- 
sistent petiole bases on the stem of Gycas revoluta. 

Fig. 1 (Plate III.) represents a section taken from Weiss' speci- 
men. Here we see the form of the leaf-bases as described by 
Weiss ; in the interior of the spaces which occur in the lower 
part of the leaf-bases are several Stigmarian rootlets with the 
central vascular bundles. These appear to be what Weiss took 
for sporangia ; that they are in reality Stigmarian rootlets I have 
not the least doubt ; they correspond exactly to those figured by 
Williamson 3 . 

As is usual with these rootlets, the peripheral cortical layer 
of parenchymatous tissue is preserved ; also the vascular bundle, 
the latter being sometimes surrounded by a delicate ring of paren- 
chymatous tissue. Anyone who has examined a number of 
sections of coal measure plants cannot fail to be familiar with 
these ubiquitous rootlets. Various observers have fallen into the 
error of mistaking these intruded rootlets of Stigmaria for tissues 
of the plants into which they happened to have bored their way 
and with which they are in no way organically connected. Prof. 
Williamson 4 refers to such mistakes made by Prof. Goeppert, who 
described in his Genres des Plantes fossiles a Stigmaria with 
bundles in the pith : the same mistake was afterwards made by 
Sir Joseph Hooker and Mr Binney 5 . Solms-Laubach 6 speaks of 
the ubiquitous nature of Stigmarian "appendices" and reproduces 
one of Renault's figures as an illustration. 

In fig. 1 (Plate III.) eight Stigmarian rootlets are seen. 

In fig. 2, the cortical tissues are shewn considerably magnified. 
At A, the section shews prosenchymatous cells more or less com- 
pressed and not very clearly defined ; as we pass along the section 
we come to other prosenchymatous cells in a better state of 
preservation, farther on these become shorter and less distinct; 
at B, the cortical tissues end : after a short break we come to the 
leaf tissue at G, where the cells appear to be parenchymatous ; as 

1 Flora der Steinkohlenformation, Fig. 33. 

2 Flora Saraeponta, PL xiv. 

3 A Monograph on the Morphology and Histology of Stigmaria Jicoides (Palse- 
ontographical Society, London, 1887), PI. x. Fig. 42. 

4 Ibid. p. 13. 

5 Q. J. G. S., Vol. xv. p. 76. 

6 Sohns-Laubach, loc. cit. p. 294. 



46 Mr A. G. Seward, On Lomatophloios macrolepidotus. [Feb. 10, 

they approach the periphery D they become somewhat oblique 
and gradually get more and more compressed. From E to F we 
have similar tissues belonging to another leaf-base. 

Williamson 1 gives a figure of Favularia shewing the " tubular 
part of the bark 2 " passing through prosenchymatous cells into 
the outer parenchymatous tissue of the epidermal layer. In my 
fig. 2, in passing from A to B, almost identically the same tissues 
are seen. From C to D and E to F we have parts of the paren- 
chymatous epidermal layer which has passed off to form the leaf- 
bases. The cells in this tissue become smaller and denser towards 
the periphery, as noticed in similar sections by Williamson 3 . 

At a in fig. 1 are some opaque bodies, round and elliptical 
in shape (too small to be shewn in the figure). These I took to 
be coprolites of some wood-boring Annelid ; such bodies are by 
no means uncommon in sections of fossil wood; in one of William- 
son's figures of a Calamite 4 some of these coprolites are seen in 
its medullary cavity, in another place 5 a piece of wood is shewn 
perforated by some xylophagous animal whose coprolites occur 
scattered about the tissues. These coprolites and the vascular 
bundles of the Stigmarian rootlets seen in transverse section 
seemed to me the only things that Weiss could have taken for 
spores. 

The conclusion to which an examination of the above sections 
led me was that this so-called cone of fructification is simply 
a flattened portion of a Lepidodendroid plant which has lost 
its woody axis, also the innermost and middle cortical tissues. 
In fig. 2, Plate xxiv. of the second of Williamson's coal plant 
memoirs we have a longitudinal section of Lepidode?idron selagi- 
noides : if we imagine all the tissues of this to be destroyed except 
those marked I and k, that is the epidermal or leaf-base tissues, 
and the tubular part of the outer bark, we have left exactly the 
same as those preserved in this specimen of Lomatophloios. 

Prof. Williamson 6 , in speaking of the bark tissues, remarks 
that near the outer surface there is a layer of prosenchymatous 
tissue where the cells are so elongated as to constitute a distinct 
"bast layer" which has exhibited a constant tendency to separate 
itself from the subjacent cortical tissue. 



1 On the Organisation of the Fossil Plants of the Coal Measures, Pt. n. Pl.xxvin. 
Fig. 32 (Phil. Trans. Boyal Society, 1872, p. 197). 

2 Ibid. p. 211. 

3 Ibid. p. 201. 

4 On the Organisation of the Fossil Plants of the Coal Measures, Pt. i. PI. xxiv. 
Fig. 10 (Phil. Trans. 1871, p. 477). 

5 On the Organisation of the Fossil Plants of the Coal Measures, Pt. x. PI. xx. 
Figs. 65, 66 (Phil. Trans. 1880, p. 493). 

e hoc. cit. Pt. ii. (Phil. Trans. 1872), pp. 223, 224. 



1890.] Mr A. C. Seward,' On Lomatophloios macrolepidotus. 47 

Further on 1 , he speaks of the very frequent occurrence, in 
remains of Lepidodendroid plants, of simply the epidermal layer 
and the semi-fibrous portion of the prosenchymatous one; all the 
inner tissues having been loosened by decay and eventually floated 
out: when the stems were prostrated, this epidermal and bast 
layer constituted the cylinder whose two sides were eventually 
brought together and flattened by superimposed pressure. In 
another place 2 , a full description is given by the same author of 
a specimen which was apparently very similar to that now under 
consideration ; the following are Prof. Williamson's words : — " I 
have before me at the present moment a section of a large Lepi- 
dodendron of which the woody axis and its medullary centre have 
disappeared, the thick cortical layer alone remaining. A large 
Stigmarian root has found its way into the cavity and filled it up, 
giving off its peculiar rootlets within the Lepidodendron cylinder. 
Such a specimen would inevitably mislead even a botanist whose 
eye was not familiar with the appearances of the two plants." 

1 Ibid. p. 228. 2 Ibid. p. 215. 

EXPLANATION OF PLATE III. 

Illustrating Mr A. 0. Seward's paper " Notes on Lomatophloios 
macrolepidotus (Goldg.)." 

Fig. 1. Section taken at right angles to the surface of Lomatophloios 
macrolepidotus, i. e. at right angles to the leaf -bases. Actual 
width of section 3*5 cm. 

a. Position of (?) Coprolites. 

b. Stigmarian rootlets. 

c. One of the leaf-bases with space containing two 

Stigmarian rootlets. 

Fig. 2. Part of the upper end of fig. 1 magnified. Corresponding 
parts shewn by the lettering in figs. 1 and 2. 

Fig. 3. Outline of the specimen (Lomatophloios macrolepidotus). 

ab = 18 cm. 
cd= 13 cm. 

The section shewn in fig. 1 is taken from that part 
indicated by **. 

Fig. 4. A few of the leaf-bases {~ natural size), shewing leaf-scars s, 
and small indentations i on the swollen leaf-bases. 

The whole of both surfaces of fig. 3 are covered with such leaf-bases. 



48 Mr Laurie, On Vehicles used by [Feb. 24, 

(3) On the origin of the embryos, in the ovicells of Cyclosto- 
matous Polyzoa. By S. F. Harmer, M.A., King's College. 

[Reprinted from the Cambridge University Reporter, February 18, 1890.] 

The species investigated belonged to the genus Crisia, in which, 
as in other forms of Cyclostomata, the mature ovicells contain a large 
number of embryos. These embryos are imbedded in the meshes 
of a nucleated protoplasmic reticulum, which also contains a mass 
of indifferent cells, produced into finger-shaped processes, the free 
ends of which are from time to time constricted off as embryos. 
The embryos have, at this stage, a structure identical with that of 
the youngest embryos described by previous authors. After de- 
veloping various organs, they escape as free larvae through the 
tubular aperture of the ovicell. The budding organ from which 
the embryos are formed makes its appearance at an early stage in 
the development of the ovicell. Evidence was brought forward to 
show that it must be regarded as an embryo, produced from an 
ovum. The supposed ovum is found in very young ovicells, im- 
bedded in a compact follicle, and appears to give rise, by a remark- 
able process of development, to the budding organ above described. 
The embryos are thus produced by the repeated fission of a primary 
embryo developed in the ordinary way from an egg. 



February 24, 1890. 
Mr J. W. Clark, President, in the Chair. 
The following were elected Fellows of the Society : 

A. P. Laurie, M.A., Fellow of King's College. 
W. H. Young, M.A., Fellow of Peterhouse. 
C. Platts, M.A., Fellow of Trinity College. 
A. C. Seward, M.A., St John's College. 

The following Communications were made to the Society : 

(1) Vehicles used by the old Masters in Painting. Part I. 
By A. P. Laurie, M.A., King's College. 

I have been engaged for some time in studying the methods 
of painting used by the old masters, with a view to showing light 
on the question why modern pictures are so far from permanent. 
Fortunately a good deal of material exists for this enquiry, MSS. 
having been left containing receipts for the preparation of oils, 
colours and varnishes, and directions for using the same. More- 
over these MSS. have been carefully edited and translated, and 



1890.] the old Masters in Painting. 49 

books have been written on the history of painting with a view to 
expounding the methods used during the best period of art. 

Unfortunately, however, in spite of the large amount of in- 
formation thus at hand, many most important points remain 
obscure. It is seldom those who best know the subject who write 
the text-books. And these MSS. are necessarily difficult to interpret 
owing to the careless unscientific spirit of the time. Weights are 
hardly ever given and times measured by Paternosters. Names 
are loosely used and are often impossible to translate, and probably 
in most cases these receipts, collected from many sources, though 
useful to artists as books of reference, do not give the actual 
methods in use in the studio. If we imagine some one trying to 
reconstruct modern industrial processes from the text-books made 
use of in technical instruction, it would help us to realize the 
difficulty of reconstructing processes of the middle ages from these 
MSS. Unfortunately also those who have written the modern 
books on this subject have been artists and archaeologists, but 
have not been chemists. Consequently whole chapters of ingenious 
learning may be devoted to proving that a certain process was used 
by the old masters, which half an hour's work in a laboratory 
would have shown to be impossible. 

I wish here to deal with the theory developed in one of these 
works, namely, Eastlake's Materials for the History of Oil Painting, 
to account for the durability of the early Flemish pictures. 

I shall assume that all that can be said from an historical point 
of view has been said by Eastlake, and that it only remains to test 
the truth of his conclusions by experiment. 

Eastlake's book is devoted practically to expounding the 
method of painting used by Van Eyck and his followers. 

His method is of special interest for three reasons : 

He may claim to be the inventor of oil painting in the same 
sense that Watt invented the steam engine. 

His pictures are remarkable for their durability. 

He was a Flemish painter, and had consequently to deal with 
a damp climate very similar to our own. His methods of painting- 
are therefore of far more value to us than the methods used in the 
dry climate of Italy. 

In considering our subject, that is the vehicle used by the 
painter, we may treat it from two points of view. 

Either considering the permanence of the vehicle itself, or con- 
sidering the capabilities of the vehicle in protecting the pigment 
mixed with it from air and moisture. 

I shall only consider here the question of the protection of the 
pigment by the vehicle. For the more we study the old methods 
of painting the more is the importance of this matter forced 
upon us. 



50 Mr Laurie, On Vehicles used by [Feb. 24, 

When I began I imagined that the old masters made use of a 
few absolutely unalterable pigments and in this way ensured the 
permanence of their pictures. I soon found this view to be 
erroneous. Many colours were described which were so fugitive 
that no modern artist would use them. Red, yellow, blue, and 
green lakes, for instance, prepared from vegetable dyes, some of a 
most fugitive character under the action of air, moisture, and 
daylight. Many of these colours may have been only used for 
illuminating parchment, where, kept from light and moisture, they 
might doubtless be permanent ; but there seems to be no reason 
to doubt that many were used in the oil painting of pictures. 
How then did these men succeed in using in oil painting colours 
known to be very fugitive and therefore avoided by modern 
artists ? 

In order to understand how this might be successfully done we 
must turn to the recent experiments of Captain Abney and Prof. 
Russell on the permanency of water colours, published in July, 
1888 (Government report). They have tested the action of sunlight 
on water colours, and they find that in dry air many pigments are 
permanent that fade in moist air, and that in vacuo hardly any 
colours are altered. If then we could ensure the absence of 
moisture and air, many of these colours now regarded as fugitive 
would doubtless be permanent. Turning again to Eastlake, we 
find that evidently the old masters quite understood this, and that 
they took especial pains to lock up fugitive colours by the intro- 
duction of some varnish ; but before going further let us put down 
briefly what vehicles were used by them for painting. They made 
use of a fine size prepared from parchment, gum arabic, white of 
egg, yolk of egg mixed with fig-tree juice and so on, but the use of 
oil was long difficult on account of its slowness in drying. It was 
soon found, however, that certain oils, such as walnut oil, poppy 
oil, and linseed oil, had the property of becoming converted into 
hard resins in time, the process technically called drying, and that 
this process could be hastened by exposing the oil before use for 
some time to the sun or by boiling it, boiled oil drying quicker 
than raw oil. It was next found that by boiling the oil with 
litharge or white lead it would dry still quicker, and was therefore 
more suitable as a vehicle for colours. It was also known that by 
dissolving in oil certain resins, such as amber, sandarac, Venice 
turpentine, and perhaps copal, and later mastic, that varnishes 
could be prepared. These facts were known before Van Eyck, and 
he had these materials from which to develope his method of 
painting in oils. There can also be no doubt that boiled oil did 
not afford a sufficient protection for fugitive colours, and that 
therefore a varnish must be ground with the colour to coat each 
particle, and so protect it from the action of air and moisture. So 



1890.] the old Masters in Painting. 51 

much has, I think, been proved by Eastlake, and we have next to 
ask what varnish was used, or would any varnish do. On this 
point it is apparently impossible to get any definite historical 
evidence on account of the looseness with which terms were used. 
Nevertheless Eastlake does come to a conclusion, and states his 
belief that an oil varnish ground with the colour is sufficient 
protection whether made from amber, copal, sandarac, or Venice 
turpentine, and discards the tradition that Van Eyck made use of 
amber varnish alone. He points out that while receipts exist in 
the MSS. for preparing amber varnish, they also exist for the 
preparation of other varnishes, and there is nothing to show that 
amber was exclusively used. In fact it is improbable that amber 
varnish alone was used, as it is difficult to prepare any which is 
not very dark in colour. So difficult in fact is it to prepare that 
it can hardly be said to be an article of commerce, though occa- 
sionally varnishes claiming to be amber are sold to artists. This 
theory of Eastlake's is easily put to the test, and he quotes an 
experiment with gamboge, by Sir Joshua Reynolds, which seems 
to confirm it. Unfortunately, however, Prof. Church has tested 
this point, and finds that Eastlake's theory is not tenable. He 
finds that gamboge fades as quickly when mixed with copal varnish 
(the best of resins now in use) as it does with oil alone. We are 
left then with this question still to solve, "What varnish did 
Van Eyck use to mix with his colours ? " 

I have made some experiments which I think point to the 
correct solution of this problem. My object has been to get some 
rapid means of deciding whether a given vehicle was or was not 
permeable to moisture, and I finally hit on the following device. 
If we ignite sulphate of copper it loses all its water of combination, 
leaving a white powder which is very hygroscopic. If this powder 
is exposed to the air for a short time it turns green, owing to the 
absorption of water. I ignited therefore some sulphate of copper, 
and using it as a pigment ground it with boiled oil, and painted 
it out on three pieces of glass. One of these I placed in a desicc- 
ator, one in a warm dry room, and one in a room with the window 
open. In 12 hours the sulphate of copper in the damp room had 
turned completely green, that in the dry room slightly green, and 
that in the desiccator remained white. I then exposed all three in 
the damp room, and they were soon equally green. This showed 
me that I had here a delicate test of the permeability of such 
mediums to moisture, though this first experiment was hardly 
fair to the boiled oil, as it had not been allowed to harden before 
exposure to moisture. For no pigment used in practice is so 
fugitive as to be affected by exposure to moisture during the time 
the oil is drying, and therefore it was obviously necessary to try 
an experiment after the vehicles had dried completely in the 

VOL. VII. PT. II. 5 



52 Dr MoncJcman, Action of Copper Zinc Couple [Feb. 24, 

desiccator. I therefore next ground my sulphate of copper with 
the following vehicles, painting each out on glass : 

(1) Boiled oil alone (best commercial), 

(2) Copal varnish alone, 

(3) Common rosin in turps, 

and two with amber varnish, one prepared by dissolving amber 
in turps, the other by dissolving it in oil. These were placed 
under a desiccator and left to dry. Some of them dried very 
slowly, and had to be assisted by warmth till at last all were dry. 
I then removed the sulphuric acid from the desiccator and replaced 
it by water, thus leaving them in an atmosphere saturated with 
moisture. Very soon some of them began to turn green, and 
after a week all had turned green except one of those painted 
with amber varnish. This experiment indicates, I think, the 
following conclusions : 

(1) That neither boiled oil nor rosin varnish, nor copal varnish, 
protect a pigment absolutely from moisture. 

(2) That amber varnish properly prepared does do so. The 
amber varnish that protected the sulphate of copper was that 
prepared with turps. The one prepared with oil failed to do so. 
Therefore we may take it that Van Eyck probably used an amber 
varnish. 



(2) On the Action of the Copper Zinc Couple on dilute solutions 
of. Nitrates and Nitrites (NaHO and KHO being absent). By 
James Monckman, D.Sc. Lond., Downing College, Cambridge. 

It is usually stated that when a dilute solution of a nitrate, 
containing a copper zinc couple, is boiled for some time, the whole 
of the nitrogen in the nitrate is given off as ammonia, which 
may be used as a means of estimating the quantity of nitric 
acid. When NaHO or KHO is added to the liquid good results 
are obtained, but when no such addition is made the quantities 
of ammonia found, by students working in the University 
Laboratory, has fallen so far short of the true amount, that it 
appeared to point to some other reaction taking place at the 
same time. I was therefore asked to examine it carefully in 
order to discover why the ammonia found was not equivalent 
to the nitrate used and what became of the remainder. 

Quantities obtained from KN0 3 . My first endeavour was to 
try what kind of results could be obtained by the method, how 
far the various experiments could be made to agree in the 
quantity of ammonia evolved and the amount of nitrogen that 
disappeared. A very large number of experiments were per- 



1890.] on dilute solutions of Nitrates and Nitrites. 53 

formed in order if possible to get some kind of regularity in the 
numbers found, but without success. 

The method employed, is that one described in text-books 
on quantitative analysis. It consists in placing a quantity of 
well washed CuZn couple in a dilute solution of the salt (KN0 3 ) 
and boiling. The steam is conducted through a condenser into 
a measured quantity of standard acid, so arranged that all the 
ammonia is absorbed. 

It is usually stated that the reaction is at an end in one 
hour, and that afterwards no further evolution of ammonia takes 
place. As this did not appear to be the case in the first set 
of observations, the boiling was continued in the succeeding ones 
until no more ammonia was given off. In doing this it was 
found necessary to evaporate to dryness, then add more water 
and evaporate again. In some cases this process was repeated 
six times before every trace of the alkaline gas was removed. 

Sometimes the reaction stopped long before the whole of the 
water passed over, and yet on the addition of more water a 
further evolution of gas took place. This was caused by the 
oxide of zinc, produced during the progress of the reduction, 
forming a covering to the copper zinc couple and thus preventing 
the access of the liquid. When more water was added, some of 
this oxide was washed off and the surface thus exposed con- 
tinued the reduction of the nitrate. 

The experiments gave numbers varying from 29 per cent, 
of the calculated quantity up to 67 per cent, when the ammonia 
had been produced slowly, while it rose from 70 to 80 per cent, 
when the boiling was urged. 

Careful search was made for insoluble or soluble nitrogen 
compounds, which might resist the action of the hydrogen, such 
as basic nitrates, but no such bodies could be detected. 

Nitrites were however found to be produced in all cases. I 
give the numbers for these experiments as an example. 

On the 1st boiling A gave 55 per cent., B gave 50, G gave 55 

2nd „ 3 „ „ 20 „ 8 

3rd „ 10 „ „ „ 9 

4th „ 7 „ „ - „ 6 

5th „ „ ZL » _°_ 

75 • 70 78 

Thus the highest number was a little more than 20 per cent, 
short of the calculated quantity. As the highest numbers were 
obtained from those solutions which had been made to boil most 
strongly, experiments were designed to try if the variation of 
the temperature caused any alteration in the chemical reaction. 

5-2 



54 Dr Monckman, Action of Copper Zinc Couple [Feb. 24, 

Effect of temperature. It was found that when the liquid 
was evaporated by heating over a water-bath no ammonia was 
produced, while the whole of the nitrate was decomposed, nitrite 
being the only substance left in the liquid. 

Action on KN0 2 . When a solution of KN0 2 was used 
instead of KN0 3 it was found that after boiling some time it 
was completely decomposed and the nitrous acid disappeared 
from the solution without giving any ammonia. 

As no nitrogen compounds could be found in the liquid and 
no ammonia was given off nor yet any acid compound of nitrogen*, 
I was driven to the conclusion that it came away as either N or 
N 2 0, probably the former, and that probably the ammonia acted 
upon the nitrous acid producing nitrite of ammonia which was 
again broken up into N and H 2 0. 

During violent ebullition excess of ammonia was produced 
and driven off, but during gentle evaporation it united with 
the nitrous acid, which was formed in sufficient quantity to 
combine with it. 

This appears more probable because, in the first experiments 
with nitrates, the nitrate was produced by the reducing action 
of the hydrogen, at the same time as the ammonia, while in the 
latter case (the nitrites) there was excess of this body from the 
very beginning of the reaction. In the first ammonia escaped, 
in the second, not. 

Methods of testing for N. In the experiments described the 
presence or absence of hydrogen gas was of no importance, but 
when it becomes a question of testing for nitrogen and estimating 
its volume, a large quantity of hydrogen becomes inconvenient. 
I first tried to avoid producing the gas, by using a solution of 
the nitrite of sufficient strength to absorb the whole of it. I 
found that it did not work well, the action being very slow 
and little gas coming away, and as it could not be more than 
a qualitative method it was abandoned. 

Next it was proposed to sweep out the air from the tubes 
by a current of gas (C0 2 ) and after boiling the solution to carry 
the gas produced into a receiver in the same manner. The 
C0 2 was to be absorbed in the usual way. The danger of 
producing a compound of NH 3 and C0 2 which might, in presence 
of the nitrite of potash, produce nitrogen after the manner of 
AmCl and KN0 2 , formed an obstacle to its use. I was therefore 
compelled to substitute H. 



* Twice I found a very slight acid reaction produced by the gas evolved and 
twice considerable quantities of NO were evolved, but after washing the couple 
until it was perfectly free from acid or acid salt, the dilute solution ceased to give 
NO. 



1890.] on dilute solutions of Nitrates and Nitrites. 55 

The method of working will be understood by the aid of the 
diagram. 




A is an apparatus for generating H which passes through 
a solution of silver nitrate in B, the nitrate of potash and the 
CuZn couple are placed in C, while D contains the acid for 
absorbing the ammonia. One vessel only is used to keep the 



56 Dr Monckman, Action of Copper Zinc Couple [Feb. 24, 

capacity of the whole as small as possible, the current of steam 
was kept very gentle and the water in the outside vessel quite 
cold, E contains the granulated oxide of copper and F the 
receiver. 

After passing the gas through B, C, D, E for 18 hours it was 
joined up to F, and sufficient gas sent through to force the water 
in the receiver down about three inches below the surface of the 
water in the outer vessel. It was then stopped and the clip G 
closed to prevent steam passing back into B. 

The liquid in C was then boiled gently for three hours, the 
quantity of gas in the receiver was prevented from becoming 
too great by carefully heating the oxide in E. Finally by a 
current of hydrogen from A the whole of the gas was swept out 
of the tubes ; as before most of this was absorbed by heating 
the oxide in E, so that a considerable quantity was passed through 
the tubes B, C, D. The clip H was next closed and the quantity 
of N in the receiver determined. 

In order to decompose any nitrate or nitrite that might 
remain in C, HKO was added to the solution and the whole 
boiled until all the salt was decomposed, after which the ammonia 
was determined. 

The results appear in the following table : 

The weight of KN0 3 in the solution was "225 grms. 

of which the N would weigh "03118 grms. 

The result of the previously described experiment was 

N evolved as gas "02304 grms. 
N evolved as NH 3 -Q09 „ 
Total -03204 



Too much by '00086 grms. 

Given in per cent, of the salt used : 

calculated, N is 13861 per cent., 
found, N is 14-24 „ 

excess, '379 „ „ 

The only other salt tried by me was the nitrate of ammonia, 
which gave nitrite on gently boiling with the couple. As that 
salt is decomposed into N and H 2 on boiling I did not consider 
it necessary to prove the evolution of N in this case. 

In conclusion I wish to say that no equation has been given, 
because the proportion of NH 3 evolved so evidently depended 
upon the temperature, or rate of boiling, that it would be mis- 
leading; the reaction may be represented however by the two 



1890.] on dilute solutions of Nitrates and Nitrites. 57 

following, combining them in varying quantities according to 
the temperature : 

(a) KN0 3 + H 8 = KHO + 2H,0 + NH, , 

08) KN0 3 + H 2 =KN0 2 + H 2 0, 

(a + /3) KN0 2 + H 2 + NH 3 = KHO + NH 4 N0 a 
NH 4 N0 2 = N 2 + 2H 2 0. 

During the course of this research I received much valuable 
advice from Mr Sell and Mr Fenton of the University Laboratory 
for which I wish to acknowledge my obligation. 

In the previous description I have mentioned the fact that 
the strong solutions of nitrates and nitrites did not produce the 
same reactions, failing to produce a sufficient quantity of gas. 

Probably this arose from the same kind of difference being 
produced in the reaction by the variation of the density of the 
solution, as by the temperature at which it took place. This 
view was not put to any test. 

(3) On certain Points specially related to Families of Curves. 
By J. Brill, M.A., St John's College. 

1. The following communication is intended as a sequel to 
my paper "On the Geometrical Interpretation of the Singular 
Points of an Equipotential System of Curves*," and in it I pro- 
pose to show that the theorems which I established with regard 
to Equipotential Systems are, with certain qualifications, true of 
a much more extensive class of systems. It is to be understood 
that the remarks contained in this paper refer to algebraical 
systems, as it is impossible to reason in a general manner about 
transcendental curves where imaginary geometry is concerned. 

In my paper entitled " Orthogonal Systems of Curves and of 
Surfaces f" I showed that if £ and rj be the parameters of two 
families of curves constituting an orthogonal system, and if h 
and h 2 be defined by the equations 

S'+®'-v 
©"♦©'-v. 

then the value of the expression 

h 2 d^ + iJ^drj 
dx + idy 

* Proc. C. P. S., vi., pp. 313—320. 
t Proc. C. P. S., vi., pp. 230—245. 



58 Mr Brill, On certain Points specially [Feb. 24, 

is independent of the ratio dy : dx. Now it is evident that there 
must be certain points or loci for which the inverse of this 
expression vanishes, and therefore the expression itself becomes 
infinite. These points or loci may be considered as a generaliza- 
tion of the singular points of equipotential systems. 

I do not propose to discuss the general case, but to confine 
myself to a large and important series of cases in which the ratio 
of the two lis is some definite function of x and y, which function 
of x and y is the same for all systems of a particular class. I 
shall write hjl\ = <f> (x, y), and instead of the expression given 
above shall consider the expression 

<f> (x, y) d% + idy 
dx + idy 

The fact that this expression has a value independent of the 
ratio dy : dx, requires the existence of the relations 

and the value of the expression may be written in either of the 
forms 

Moreover it follows that the value of the expression 

dx + idy 
cp (x, y) d% + idy 

is independent of the value of the ratio dg : dn, and this requires 
the existence of the relations 

dx , , .dy , dy , , .dx 

^ = <l>(x,y) f v and f^-t&y)^, 

and its value may be written in either of the forms 

dy .dx 1 (dx . dy 

dn % dn 4>(x,y)\di; 9f 

It is also to be remarked that the relations given above lead 
to the equation 

dx dx dy dy 
d% dn di; dy 



1890.] related to Families of Curves. 59 

Thus, if ds be the length of an elementary arc, we have 
ds' = d,' + dy> = gdf + f v d v J + {|df + |*,f 

2. We proceed to discuss the properties of the loci of ulti- 
mate intersections of the two families of the orthogonal system. 
Suppose that P is the point (x, y), and that Q is a neighbouring 
point on the curve of the family r\ which passes through P ; then 
by the preceding article we have 



*M©'+®1* 



v 



If this vanish independently of the value of d%, then two con- 
secutive curves of the family £ cut at the point P. The vanishing 
of the said expression requires that either 

dx = dy_ 

dec , .dy _ 

The first of these conditions has reference to the existence of 
a real locus of ultimate intersections, as is otherwise evident. The 
equations 

--0 and ^-0 
8 £-U and af _u, 

if expressed in terms of x and y f would denote two curves which 
would in general intersect in a set of discrete points. If however 
the expressions dx/dg and dy/dg had a common factor, then two 
branches, one belonging to each curve, would coincide, and con- 
sequently the curves of the family £ would have a locus of ultimate 
intersections not consisting wholly of discrete points. We can 
however show that there are restrictions on the form of such a 
factor should it exist ; for if this factor be not such that if equated 
to zero it secures that cf> (x, y) shall at the same time vanish, then 
it follows that at all points for which 

^ = and A=0. 

. 01; 0% 

We have also 

3 / = and 9-0. 

07) 07) 



60 Mr Brill, On certain Points specially [Feb. 24, 

Thus at all points of the locus of ultimate intersections we have 
dx = ~7.d%+ ~- clr] = 

and dij = ^Ld^ + ^ k drj = 0. 

These equations give x = const, and y = const., and thus the locus 
of ultimate intersections consists of a set of discrete points. 
Moreover this set of points is such that the coordinates of each 
of them make the expression with which we started infinite. 

Thus we have, in general, that if two consecutive members of 
one of the two families intersect, then all their points of inter- 
section are points of the character under discussion, and it follows 
that all the members of the family pass through this set of points. 
The only exception to this is that the curves of the £ family may 
possibly have for a locus of ultimate intersections the locus of 
the points for which </> (x, y) vanishes, and the curves of the rj 
family may possibly have for a locus of ultimate intersections the 
locus of the points for which the same expression becomes infinite. 

The question now arises whether these special loci, should 
they exist, belong to the generalization of the singular points of 
an equipotential system. This depends on how we define that 
generalization. If we define it with the aid of our original ex- 
pression, we see that these loci do not belong to it if they arise 
from the vanishing of 1\ or h 2 , but if they take their origin from 
either of these quantities becoming infinite then they do so 
belong. If on the other hand we define it with the aid of the 
expression 

d£ , .dr> 

Jh A 

dx + idy ' 

then the exact contrary is the case. 

It does not follow that either of the two families should be 
such that all its members pass through the points under dis- 
cussion, although it is true that if any two belonging to one 
family do so, all belonging to that family will. Also it is evident, 
on account of the orthogonal property, that if one of _ the families 
be such that all its members pass through the said system of 
points, then those of the other family do not do so, only one 
member of that family passing through any one of the points. 
It is however conceivable that there may be cases in which the 
members of one family pass through some of the points, and the 
members of the other family through others of them. 



1890.] related to Families of Curves. 61 

3. We now pass on to discuss the case of imaginary loci of 
ultimate intersections. These are given by the condition 

It will only be necessary to consider one of these two cases, 
as the discussion of the two will be exactly similar, and we will 
choose the one given by the upper sign. The relation 

dx . dy _ - 
requires also the existence of the relation 

drj dri 

dx . dy n 
i.e. — + t ^ = 0, 

07] 07] 

unless the first of these relations makes <f) (x, y) zero or infinite. 
Thus, with this reservation, we have at all points of an imaginary 
locus of ultimate intersections 

dx + idy = J^d% + ^d7] + i(^d!; + ^d7])=Q, 

i.e. x + iy = const. 

If we had taken the lower sign we should have had 

x — iy = const. 

Thus we see that if there be any imaginary locus such that 
the coordinates of all points on it make (j> (x, y) zero or infinite, 
then it is conceivable that this may be part of the locus of 
ultimate intersections of one of the families of the system. If 
not, then the imaginary loci of ultimate intersections, whensoever 
they exist, are collections of straight lines passing through the 
circular points at infinity. Each of these imaginary straight lines 
will pass through one real point, and these points will belong to 
the set of points characterized above. Further, as we are only 
considering algebraical curves, for any imaginary point of inter- 
section {a + ib, c + id) there will be a conjugate point of inter- 
section (a - ib, c — id). Thus it is evident from the reasoning 
contained in my former paper that the imaginary straight lines 
occur in pairs, the two members of each pair being conjugate and 
intersecting in a real point, which is one of the points in question. 

The reasoning of this article proves only that if imaginary 
loci of ultimate intersection exist, then with the restriction speci- 



62 Mr Brill, On certain Points specially [Feb. 24, 

tied above they are of the character described. Every locus of 
the form x ±iy = const, will cause the expression for ds to vanish, 
but it is only such of these loci that pass through the points 
under discussion that form part of the locus of ultimate inter- 
section of the members of either of the families. 

4. As the demonstration of the preceding article is attended 
with some difficulties, it will be perhaps well to give another 
demonstration modelled on one given by Kummer in the paper 
referred to in the communication to which this is a sequel. If 
we write 

£ + n = 2m and f 17 = v 2 , 

then we may consider the parameters of our two families of curves 
as the roots of the equation 

a 2 - 2aw + v 2 = 0. 

To find the loci of ultimate intersections of the curves given by 
this equation, we have 

a — u = ; 

and the loci we are in quest of are given by 

w a = v 2 , 

or u = + v. 

We will now discuss the direction of the tangent of one of 
these loci. We have 

du du dy _ \dv dv dy\ 
dec dy dx ~ \dx dy dx) ' 

du dv 
dy _ dx~ dx 
dx du dv ' 

dy-dy 

It remains to express the condition of orthogonality of the two 
families of curves. We have the equations 

3| dt) _ du 

dx dx dx ' 

dx dx dx' 

and from these it follows that 

^-^dx =2 \Zdx- v dx)' ^-^ = - 2 r^~^r 



= 0, 



1890.] related to Families of Curves. 63 

Similarly we should obtain 

/*. n^ o ( fdu dv) , y ,dr) f du dv) 

Substituting these values in the equation 

9f dr) + dj[dv = 

dx dx dy dy ' 
we have 

( <,du dv\ ( du dv\ /f.du dv\ ( du dv 
{^x- V ^)[ V ^- V dx) + ^Fy- V dy)[ 7) dy- V d] / 
or 

fdu\* /dv\* (fa\* + fiv Yl_9 i- t9 - ?^ — 
^W \<W \dyj W/ I 1.3# 3# ty ty 

Now if w = ± v, this equation becomes 

/duV /dv\* / a l t V + ^V+2i-- ?^-l-0 
V3ay \3ay V3t// \3y/ ~ \dx dx dy dy\ ~ ' 

\du dv) 2 {du dv) 2 n 

i£±5f + V5r* 

du dv 

dx ~ dx 

or s s- = + %. 

du dv 

dy-dy 

This proves our point. This proof is not identical with Rum- 
mer's, though modelled on it*. The mistake made by Kummer 
was to assume that the locus of ultimate intersections necessarily 
constituted a proper envelope, and thus that the points under 
discussion were necessarily foci. I have however dwelt at sufficient 
length upon this point in my former paper, and consequently it 
will need no further discussion here. 

5. It now only remains to point out a few examples of classes 
of systems that come under the case we have been discussing. 
Our first example will be drawn from systems of curves which 
furnish solutions of the equation 

d 2 u d 2 u _ 2 
dx 1+ dy 2 ~ CU ' 

* If we express the equation' v?=v 2 in terms of % and -q it becomes £ 2 + rf + £17 = 0, 
and the loci of ultimate intersections would therefore appear to be wholly imaginary 
(exception being made of the points spoken of above). Kummer's method of 
investigation therefore gives rise to the question whether, if a family of algebraical 
curves have an envelope which is not a set of discrete points, their orthogonal 
trajectories will necessarily consist of a family of transcendental curves. 



64 Mr Brill, On Families of Curves. [Feb. 24, 

Suppose that we have two functions u and v, which are con- 
nected by the relations 

du dv , du dv 

~ 5- =cu and 5- + 5- = — cv. 

ox oy oy ox 

From these we easily deduce 

{fx + i dl)^ + iv) = c{u - iv) ' 



dy 

,d_ 
\dx ". dy 



% ^-)(u — iv) = c(u + iv). 



Thus we have 



g^2 + ^1 (w + iv) = c 2 (u + iv), 



and therefore u and v are both solutions of the equation given 
above. 

Further, the relations connecting u and v may be written in 
the form 

These relations show that we may express u and v in terms of 
two new functions <j> and i/r as follows : 

u = <rf, v = e^, u = -e-^, v = e- d f. 
oy ox ox oy 

From this it follows that 

2cx d_±_d± d ^±__d± 
ox oy oy ox 

Another good example is given by cases of irrotational fluid 
motion symmetrical with respect to an axis. In this case we 
have a velocity potential (f> and a stream function ty connected. by 
the relations 

dr dz ' dz dr ' 

These two examples will suffice to show that among the classes 
of systems discussed in this paper there are some which have 
applications to problems of interest. 



1890.] Mr Gardiner, On Acacia sphaerocephala. 65 

March 10, 1890. 
Mr J. W. Clark, President, in the Chair. 

The following Communications were made to the Society : 

(1) On the germination of Acacia sphaerocephala. By 
W. Gardiner, M.A., Clare College. 

[Reprinted from the Cambridge University Reporter, March 18, 1890.] 

Seeds of this well-known myrmecophilous plant have been 
lately germinated, both at Kew and Cambridge, affording oppor- 
tunity of observing seedlings in all stages of growth. The striking 
manner in which this plant exhibits definite structural adaptations 
for the benefit of the ant colonies, which so efficiently garrison it, 
has suggested that the structures in question possibly owe their 
formation to the action of the ants themselves, and arise in con- 
sequence of local stimulation produced by biting (when in search 
of sweet sap) or even by stinging. Actual observation shows that 
as the seedlings assume the adult foliage, the stipular thorns, the 
petiolar glands, and the "food bodies" all develoj>e quite normally, 
and in spite of the ants being absent. It is clear therefore that 
in plants which exist at the present time the structures appear 
without the intervention of ants. It may, however, be urged that 
they were in the first instance brought into existence by ants, 
through the stimulation of ancestral forms, such stimulation 
having been persistent and extending over many generations. 
A study of germinating seedlings does not appear to support this 
view. The development of the several structures can be traced 
so gradually and through so many transition forms that there 
appear to be strong grounds for believing that the whole of the 
complicated arrangements and of the several organs concerned 
are the outcome of variation, and that the ants have done little 
more than take advantage of the results of such variation. More- 
over no absolutely new organs are present, the thorns being as 
well, or even better, developed in other species of Acacia; e.g. 
Acacia latronum: the petiolar nectaries being well known and 
common structures : and the food bodies (as their development 
shows) being hypertrophied " Reinke's glands." 

(2) Additional note on the thickening of the stem in the 
Cucurbitacege. By M. C. Potter, M.A., St Peter's College. 

Hitherto very few Dicotyledons have been described which 

possess a ring of normal collateral bundles and whose stem at 

the same time does not increase in thickness by means of a 

cambial ring. De Bary 1 mentions under this category the Saururew 

1 Comp. Anat. Eng. Ed. p. 454. 



66 Mr Potter, Additional note on the thickening [Mar. 10, 

and some species of Ranunculus ; and to this short list we may 
add some water plants, e.g. Hippuris and Myriophyllum, etc. 

The reason why so few Dicotyledons do not increase in 
thickness may be sought for in the habit and mode of life of 
this important series of the vegetable kingdom. Many Dicoty- 
ledons are perennial, being trees or shrubs, and require, as they 
grow older and larger, that additional phloem and xylem should 
be continually formed both for purposes of nutrition and also for 
mechanical support ; and this is accomplished by means of the 
cambium ring which adds continually new phloem and xylem to 
the vascular bundles. Among the herbaceous Dicotyledons in- 
crease in thickness is also the rule ; for these plants having annual 
stems require that sufficient phloem and xylem should be made 
for purposes of nutrition before the bundles are closed, and that the 
mechanically supporting tissue should be sufficiently strengthened 
to bear the increasing strains as the plant grows larger ; and hence 
a cambium ring is necessary for nutrition and mechanical support. 

The few Dicotyledons mentioned above live under special 
conditions ; — Hippuris, Myriophyllum, being water plants, require 
little xylem or supporting tissue ; and hence continual additions 
to the xylem are not needed, and so the bundles are closed. Again 
the Saururece and the species of the Ranunculacece (viz. Caltha 
palustris and species of Ranunculus) are marsh plants, which do 
not attain to any great size and whose habit is very similar to 
that of water plants ; and hence these do not require the continual 
addition of xylem and phloem to their vascular bundles. 

The Gucurbitaceod may roughly be divided into (1) herbaceous 
climbers with annual stems and (2) woody perennial climbers. 
The former on the one hand have the stem strengthened by a ring 
of sclerenchymatous tissue situated in the cortical tissue between 
the epidermis and the vascular bundles. The support derived 
from this sclerenchyma and the xylem which is formed before the 
bundles are closed, together with that which is derived from the 
external object upon which the plant climbs are sufficient; and 
hence additional xylem is not required for purposes of support, 
the plant only requiring that the amount necessary for nutritive 
purposes should be made before the bundle is closed. The latter 
on the other hand have no ring of sclerenchyma, and derive their 
support from the xylem and external objects; but since these 
plants are perennial, and continually make fresh leaves and 
branches, they require the constant addition of new xylem and 
phloem to their stems ; and from this fact we have the reason why 
a cambium ring is present. 

The explanation why an additional layer of phloem on the 
inside of the xylem is needed by the members of this order must 
again be sought for in the special conditions under which they 



1890.] of the stem in the Cucurbitacece. 67 

live. We have seen that in the herbaceous species a large amount 
of xylem is not needed ; but, by comparison with other herbaceous 
plants, we see that a considerable amount of phloem is necessary. 
This phloem receives considerable additions before the bundles are 
closed, and we may infer that it is more beneficial that the phloem 
should be divided by the xylem than that it should be placed only 
on the exterior of the xylem. 

In the woody perennial species, the amount of internal phloem 
is not large ; and the requisite amount is formed in the normal 
way by the cambium as growth in thickness takes place. These 
stems have no sclerenchymatous ring, and hence derive their 
support from the xylem and external objects, and hence, as the 
tree grows larger, continual additions are made to the vascular 
bundles. 

[Received March 9, 1890.] 



(3) Note on the action of Rennin and Fibrin-ferment. By 
A. S. Lea, Sc.D., Caius College, and W. L. Dickinson, Caius 
College. 

[Reprinted from the Cambridge University Reporter, March 18, 1890.] 

The experiments which were demonstrated to the Society 
were made in order to verify some recent statements of Fick 
{Pflugers Arch., Bd. 45, 1889, S. 293). This observer had urged 
that rennin certainly, and fibrin-ferment probably, produced their 
respective actions on milk and blood-plasma without the necessary 
contact, throughout the whole fluid mass, of the ferment molecules 
with the molecules of casein and fibrinogen. If this were so, then 
the mode of action of these ferments would be strikingly different 
from that of the ordinary digestive ferments. Fick based his 
views upon the result of an experiment made by placing a 
glycerine extract of rennin at the bottom of a tube, carefully 
pouring some milk on the top of the glycerine and observing that, 
without mixing the two fluids, the milk rapidly clotted throughout 
its entire mass. 

The authors experimented by warming milk to 40° C. in a 
test-tube, and then carefully introducing, by means of a fine glass 
tube, an active extract of rennin below the milk. The constant 
result of this experiment was that a clot was rapidly formed at 
the junction of the two fluids, but that not until the lapse of 
several hours was the milk clotted throughout. When a similar 
experiment was made with dilute salt-plasma a narrow clot of 
fibrin was similarly formed at the junction of the plasma and 
fibrin- ferment solution. But unlike the case with milk, the super- 

VOL. VII. PT. II. 6 



68 Mr Bateson, On Egyptian Mummied Gats. [Mav. 10, 

jacent plasma was never clotted up to its surface even after 24 hours' 
digestion at 40°. When however at the end of this time the fer- 
ment and the plasma were mixed by shaking the tube, clotting 
throughout the whole mass speedily occurred. From these experi- 
ments the authors concluded that Fick's views are not tenable, 
and that there is no reason for supposing that the mode of action 
of these ferments differs essentially from that of other ferments. 
They explained the results obtained by Fick and themselves in 
the case of milk as due to the inevitable mixing of traces of the 
rennin with the milk, and pointed out how slight an amount of 
mixing would suffice to produce the observed result by calling 
attention to the fact that rennin will clot 400,000 — 800,000 times 
its own weight of casein. Their results obtained with dilute salt- 
plasma were even more striking, since with this the clot never 
extended, even after prolonged digestion, more than a few milli- 
meters above the junction of the surfaces of the ferment solution 
and plasma. 



(4) On some Skidls of Egyptian Mummied Cats. By W. Bate- 
son, M.A., St. John's College. 

[Reprinted from the Cambridge University Reporter, March 18, 1890.] 

Six skulls and two restored heads of Egyptian mummy-cats 
were shown in illustration of the early history of the domestication 
of the cat. The specimens indicate that the cats embalmed by 
the Egyptians were of at least two kinds, and that the larger 
variety was of much greater size than that usually reached by 
either the modern domestic cat or the wild cat of Europe. These 
facts have been already pointed out by de Blainville and Nehring, 
but on comparison with a series of modern skulls it is not possible 
to support the attempt to refer these animals to any particular 
species of cat. The presumption is rather that cats of many kinds 
and sizes, possibly distinct, and probably including Felis serval 
and F. caligata (? = F. maniadata and F. caffra), were all thus 
embalmed ; but whether these animals were all domesticated or 
whether some were merely collected from time to time there is 
no evidence to show. 

Pupa-cases of the maggots which had lived in these heads 
were also exhibited. 



1890.] Mr Larmor, Of Electrification on Ripples. 69 



April 28, 1890. 
Mr J. W. Clark, President, in the Chair. 

The following Communications were made to the Society : 

(1) On the series in tukich the exponents of the powers are the 
pentagonal numbers. By Dr Glaisher. 

The principal result was that the square of the series 

l + ? + r/ + g 5 + r/ + 9 12 + c/ 5 + 2 22 + r/ 6 + &c. 

(which is known to be equal to the product 

(1 + q) (1 + cf) (1 - cf) (1 + f) (1 + cf) (1 - cf) (1 + cf). . .) 

is the series 

1 + E (13) q + E (25) cf + E (37) cf + E (49) cf + &c. 

where E (n) denotes the excess of the number of divisors of n 
which = 1, mod 4, over the number which = 3, mod 4. 



(2) The Influence of Electrification on Ripples. By J. Larmor, 
M.A., St John's College. 

The relation between the period and the wave length of ripples 
on the surface of a liquid must obviously be sensibly affected by an 
electric charge communicated to the surface. 

To investigate the amount of the effect, let us take the origin 
of coordinates on the surface, the axis of y vertically downwards, 
and the axis of x along the direction of propagation of the ripples. 

A suitable form for the electric j)otential V above the liquid is 

V = - Ay + ACe~ my cos mx. 

The equation of the surface (V= 0) is then 

y = G cos nix, ' 

and the surface density a of the electric charge is equal to A^ir. 

The velocity potential of the wave motion is a function <j) 
which satisfies 

dx*^dif~ ' 
and gives at the surface (y = 0), 



deb dO 

-v- = -j- cos nix. 

ay at 



G— 2 



70 Mr Larmor, On the Influence [April 28, 

Thus when the depth is so great that the ripples do not disturb 
the bottom 

1 dC .. 

<b = =- e cos mx, 

T m at 

as this leads to zero velocity at a great depth. 

dV 

The electric charge diminishes the surface pressure by — a -^— , 

— ( o— ) > or approximately j— ( -=- j , which is equal to 
(1 + 26m cos mx). 



or 

4"7r V On J 

A 



4tir 
The surface tension T diminishes the surface pressure by 

d 2 u 
T ■—-, , or — TCm* cos mx. 

dx 2 

Now the equation of fluid pressure is 

£= const. +gy-^ -h v ; 



so that we must have at the surface, approximately, 
— - — (1 + 26m cos ??z#) H cos ma? 

47Tp p 

1 cf 6 
= const. + qu cos m« ^ cos ma?, 



which requires 



d 2 „m 2 /_. go A\ 
ctr p V ^ 2777 



This gives the periodic time 

m ' / V m 27r 



A,p 27T p 

where A, is the wave length ; and the velocity of propagation is 
277-T g\_87ra 2 Y 



\p 2tt p 

The result is in fact the same as would be produced by a decrease 
in surface tension of amount 

8TT(i 2 /m, or 4o- 2 X,. 



1890.] of Electrification on Ripples. 71 

This quantity depends on X, as might have been expected, for 
the mechanical effects of an electric charge on the surface cannot be 
represented as a diminution of surface tension. To produce a 
simple reduction of tension electrically we must have the double 
condenser larger than has been assigned by von Helmholtz as the 
cause of voltaic polarization. 

When, as in the case of voltaic polarization, the ripples occur at 
the interface between two liquids of densities p x and p 2 the above 
formulas will clearly be applicable on substitution of p x + p 2 for p 
andgfa-pj/fa + pjfarg. 

The actual values here deduced come from the form of <jf> that 
belongs to fluid of some depth compared with X ; but it is obvious 
that the surface tension effect combines with the electric effect in 
the same way in every case, and that the statement just made 
holds generally. 

As the length of the ripples diminishes, the effect of the electri- 
fication is ultimately negligible compared with that of the surface 
tension, though it persists much longer than the influence of 
gravity. 

In the special case considered above the period becomes 
imaginary if X lie between the values 




so that, if a can be made so great that these limits are real, the 

wave lengths that lie between them cannot exist. For a given 

period there will be a wave length above these limits for which 

gravity is chiefly operative, and one below them for which surface 

tension is chiefly operative. 

To obtain a rough numerical estimate : On a circular plate of 

radius a changed to potential V the electric density at distance r 

i 
from the centre is a = F/7r 2 (a 2 — r 2 )' 2 , while on a sphere of the same 

radius the electric density is er = Vj^ira, which is rather less than 
that at the centre of the plate. At the centre of the plate the 
effective diminution of surface tension will be 4 V 2 X/7r 4 a\ If we 
take a 10 cm. and X 1 cm. this gives about F 2 /2500 in C.G.S. units. 
The value 33 for V makes the striking distance in air between 
balls 2 cm. in diameter about '3 cm., while according to Mascart the 
value 400 makes the striking distance between balls 22 cm. in 
diameter about 10 cm.; the former value gives an effective diminu- 
tion of surface tension of \\, the latter gives 64. For water the 
actual surface tension is about 80, for mercury 540. The electric 
effect is therefore considerable : thus Prof. C. Michie Smith (Proc. 
R. S. Edin., March, 1890) has observed an effective diminution of 
20 per cent, in the case of mercury owing to electrification. 



72 Mr Love, On Sir William Thomsons estimate [April 28, 

(3) On Sir William Thomson's estimate of the Rigidity of the 
Earth. By A. E. H. Love, M.A., St John's College. 

(Abstract.) 

In Thomson and Tait's Natural Philosophy (part II. art. 834 sq.) 
there is given an estimate of the rigidity of the earth derived from 
a consideration of tidal phenomena. For the purpose of obtaining 
such an estimate the earth is regarded as a homogeneous elastic 
solid sphere, of gravitating matter and incompressible, which is 
supposed to be strained by its own gravitation and by the action of 
external disturbing bodies such as the moon. The amount of the 
tidal distortion is measured by the ratio of the spherical harmonic 
deviation of the disturbed surface from the mean spherical figure 
to the radius of the mean sphere, and this is compared with the 
corresponding ratio in case the rigidity is annulled, i.e. in the case 
of the earth regarded as a homogeneous fluid sphere disturbed by 
the same system of forces. We may for shortness speak of the first 
dynamical system as a ' solid earth,' and of the second as a ' fluid 
earth.' It is shown that the ratio of the amount of tidal distortion 
in a * solid earth ' of the same rigidity as glass to that in a ' fluid 
earth 5 is about "612 or nearly f, while if the rigidity be taken to 
be that of steel the ratio is about "321 or nearly i In a perfectly 
rigid 'solid earth' the ratio would be zero. The height of the tide 
as given by the observable rise and fall of the sea relatively to the 
land is the difference of the spherical harmonic deviations of the 
' fluid earth ' and the ' solid earth ' by which the actual earth can 
be most approximately replaced. It is concluded that in case the 
rigidity were that of steel the height of the tide would be reduced 
to about | of what it would be in case the earth were perfectly 
rigid, and that if the rigidity were that of glass the height of the 
tide would be reduced to about § of what it would be in case the 
earth were perfectly rigid. From the observations that have been 
made Professor G. H. Darwin, whose method and conclusions are 
given in the same volume, deduces that the actual amount of the 
observable fortnightly tide cannot be much less than §-, and cer- 
tainly cannot be nearly so small as f of that calculated in the 
ordinary equilibrium theory in case the earth is regarded as per- 
fectly rigid, thus confirming Sir William Thomson's estimate of the 
' tidal effective rigidity ' of the earth, that it is much greater than 
that of glass and probably about that of steel. 

It appeared to me to be worth while to try to find out what 
would be the effect of supposing the material of the solid replacing 
the earth to have finite compressibility as well as rigidity. For 
most solids that have been tested by experiment the two elastic 
constants, the resistance to compression, k, and the resistance to 
distortion, n, are connected very nearly by the relation 3k — bn, so 



1890.] of the Rigidity of the Earth. 73 

that in replacing the earth by an elastic solid mass of gravitating 
matter we may perhaps be enabled to estimate better the amount 
of the tidal distortion if we assume this relation to hold. I have 
thei-efore solved the following problem — A mass of solid matter, 
homogeneous in the natural state, free from all applied forces, and 
filling a spherical surface, being given, such solid is strained by its 
own gravitation according to the Newtonian law, and by the 
application of external disturbing forces having a potential ex- 
pressible in spherical harmonic series. Supposing the deformed 
free surface expressed as a harmonic spheroid, it is required to find 
the amount of the harmonic inequality. — The notation of Thomson 
and Tait's Natural Philosophy is used and the solutions of the 
parts of this problem there considered are adopted. The problem 
has not been previously solved with the generality here considered. 
Now the general equations of equilibrium of the solid are 
three, of the type 



in 



doc 



+ /,V 2 a + pX = (a), 



and it is well known that the solution consists of two parts, (1) any 
set of particular integrals of these equations, and (2) such comple- 
mentary functions satisfying a system identical with (a) when the 
terms depending on the bodily forces such as X are left out as 
with the set of particular integrals (1) will satisfy the condition 
that the external surface remains free from stress after the defor- 
mation. The complementary functions for our problem are given 
in Thomson and Tait, Art. 736, and the method here used for 
obtaining the particular integrals is the same as that of their 
Art. 834, but the surface conditions cannot be immediately written 
down by their method. This happens for two reasons — firstly, 
because the stress arising from the attraction of the mass is so 
great compared with the other stresses, that its amount has to be 
estimated at the external deformed surface and not at the mean 
spherical surface as the others may be, and secondly, because part 
of the system of bodily forces consists in the attraction of the har- 
monic inequalities whose expression involves the complementary 
functions. It is however easy to surmount the latter difficulty by 
forming an equation giving the potential of the harmonic inequali- 
ties in terms of the disturbing potential and quantities that occur 
in the expression of the complementary functions. The method of 
estimating the surface tractions that must be regarded as applied 
to the mean sphere in consequence of the attraction of the mass, I 
have considered in a previous paper (Proc. Lond. Math. Soc, Xix. 
pp. 185 sq.) in the case of vibrations, and a like method applies 
here. The total surface traction arising from complementary 
functions and particular integrals is thus found and resolved into 



74 Air Love, On Sir William Thomsons estimate [April 28, 

three components parallel to the coordinate axes, and each of these 
is equated to zero. The result is the determination of all the un- 
known harmonics occurring in the complementary functions, i.e. 
the complete solution of the problem, and in particular an ex- 
pression is obtained for the amount of the harmonic inequality. 

It is of some importance to notice the way in which gravity 
rigidity, and compressibility, occur in the result, and I shall for 
simplicity of statement here limit the expression of the results to 
two cases. In the first, which is that considered by Thomson and 
Tait, the matter is supposed incompressible. In the second the 
matter is supposed compressible as well as rigid in such a way that 
the constants m and n are connected by the relation m = 2n, equi- 
valent to the relation 3k = 5n above referred to. In both the 
density is supposed equal to the earth's mean density, and the dis- 
turbing forces derivable from a potential, which is a spherical solid 
harmonic of order 2, say W 2 . This is the case for tidal disturbing 
forces. Then in both cases the amount of the harmonic inequality 
is expressible in the form eWJg where g is the value of gravity at 
the surface, and the number e is a rational function of a certain 
number ^ such that (3^) _i is the ratio of the velocity of waves of 
distortion in the material to the velocity due to falling through a 
height equal to half the radius of the sphere under gravity kept 
constant and equal to that at the surface. In the first case I find 

15fr (K . 

agreeing with Thomson and Tait's result, and in the second case 

S 3356500 + 863100S + 55485S- 8 ( , 

6 ~ 70 + 9S- ' 53900 + 27160* + 2601*' '"^' 

The value of * is zero when the matter is perfectly rigid, about 
| when the rigidity is that of steel, about 5 when the rigidity is 
that of glass, and infinite when there is no rigidity at all. 

We may regard e as an ordinate and * as an abscissa and 
trace the curves (b) and (c). The curve (6) is a rectangular hy- 
perbola, and the asymptote e = constant gives the limiting value f 
of e when the rigidity vanishes. The branch of the curve (c) which 
passes through the origin lies very near to the corresponding branch 
of the curve (b) for all positive values of S-. It has an asymptote 
giving the limiting value of e something less than f. For all values 
of S- lying between S- = and * = 5, the curve (6) lies below the 
curve (c) but the distance is very minute. For some value greater 
than S- = 5 the curve (b) crosses the curve (c) and the value of e 
given by supposing the matter incompressible is greater than that 
given by supposing 3k = 5n. The value of e for steel (* = §) is 
about '803 as given by the curve (b) and about '856 as given by (c), 



1890.] of the Rigidity of the Earth. 75 

and for glass (^ = 5) the value of e is about T53 as given by (6) 
and about 154 as given by (c). To compare the heights of the 
tides in these cases with those given by the ordinary equilibrium 
theory we have to multiply by §, the result is the fraction by 
which the height of the tide is reduced by the elastic solid yielding. 
It is as in the case considered by Thomson and Tait about i when 
the rigidity is that of steel, and about § when the rigidity is that of 
glass. 

May 12, 1890. 

Mr J, W. Clark, President, in the Chair. 

The following communications were made : 

(1) The action of Nicotin upon the Fresh-iuater Crayfish. By 
J. N. Langley, M.A., F.R.S., Trinity College. 

When 1 to 3 mgs. of nicotin, in a 1 p. c. solution, are injected 
beneath the epidermis of a crayfish, there is a very remarkable 
paralysis of certain functions of the nervous system. Almost 
immediately after the injection there is a tetanic contraction of 
the striated muscles of the body, so that the eye-stalks are drawn 
in, the antennae and antennules bent backwards, the ambulatory 
legs are flexed, the claws firmly closed, and the tail bent. This 
tetanic condition lasts for a minute or two only. Before it passes 
off, or a little after it has done so, there is a strong peristalsis of 
the intestine with a rhythmic movement of the vent; the duration 
of this is variable, it may go on for an hour or more ; for a time 
after it has ceased the movement of the vent can be readily pro- 
duced by local stimulation. But neither the tetanic contraction of 
the striated muscles nor the rhythmic movement of the unstriated 
muscles are effects peculiar to nicotin ; many other substances have 
a similar action. 

After the tetanic period, there follows a period of complete 
fiaccidity of all the striated muscles, the respiratory movements 
have ceased, and there is no movement of any segment; reflex 
movements are for a short time abolished, but soon slight sluggish 
reflex movements of flexion or extension may be obtained from any 
one of the ambulatory legs by scratching the shell of the leg; the 
movement is usually local, though sometimes the segment above 
and the segment below also move. 

In a quarter of an hour or more, — the time varying with the 
amount of nicotin given and with the condition of the crayfish — 
the normal movements of the scaphognathites begin again, about 
the same time the flagella start their active, lashing movements, and, 
though as a rule somewhat later, there are fairly normal rhythmic 
movements of the swimmerets. Each of the three structures 



76 Mr J. N. Langley, The action of Nicotin [May 12, 

mentioned has at irregular intervals periods of rest, these are most 
frequent and longest in the case of the swimmerets ; least frequent 
and shortest in the case of the scaphognathites. The movements 
can be stopped for a few seconds by various slight sensory stimuli. 

Although the scaphognathites, fiagella, and swimmerets thus 
rapidly recover their ordinary rhythmic action, the crayfish lies as 
it is placed and there is an almost complete absence of other 
movement; there may be a slight slow movement of an ambulatory 
leg, but this is probably of a reflex nature. This condition is, in 
its chief features, maintained for one to two months. During this 
time there is a progressive slight improvement in the reflexes so 
that stimulation of any one segment causes a greater local effect 
and gives rise more readily to movements in other segments. The 
reflex movement however never becomes very active. The last seg- 
mental reflex to recover is the closure of the great chelae on touch- 
ing their inner surfaces ; there are some other odd points in the 
order and extent of the recovery of the reflexes, but as these have 
not been quite the same in all cases, I shall consider them at 
a later time. If food is pushed into the oesophagus, it is apparently 
carried on and digested, but I have not seen any recovery in the 
power of the maxillae or mandibles to aid in taking up food; in one 
case only (five to six weeks after nicotin injection) did the chelae 
of the first two ambulatory legs close on placing a small piece of 
food between them, but even then no movement was made to carry 
the food to the mouth. 

Two months is the longest time for which I have kept a cray- 
fish after it has received 2 or 3 mgs. of nicotin ; in every case there 
was some cause, such as the cessation of the water supply, to which 
death may have been due. I am, then, unable to say whether or 
not the above-mentioned amount of nicotin is eventually fatal. In 
the paralysed state of the crayfish, fungus rapidly grows on the 
animal, and this unless frequently removed is sufficient to cause 
death. 

The paralysing action of nicotin upon the crayfish is due to its 
affecting the central nervous system, and not to its affecting either 
the peripheral nerves, the nerve-endings, or the muscles. This can 
be shown readily by giving nicotin to a crayfish and then stimu- 
lating one of the peripheral nerves, the stimulation causes contrac- 
tion in the muscles of that segment. 

It is clear also that the action of nicotin on the central nervous 
system is to an extraordinary degree selective, that is to say, that 
certain parts of the central nervous system are affected very much 
more than others; thus the nervous action causing rhythmic move- 
ments of the scaphognathites, fiagella and swimmerets soon 
returns to its normal condition, whilst the nervous action lead- 
ing to walking, swimming, masticatory movements, etc. are stopped 



1890.] upon, the Fresh-water Crayfish. 77 

for a mouth or two if not permanently. Broadly speaking those 
actions which may be considered voluntary are the most interfered 
with by nicotin. 

Further, a comparison of the effects of nicotin on the crayfish 
and of the effects of section of different parts of the central nervous 
strand, such as those made by Ward, makes it probable that in each 
ganglion there is a part which is paralysed by nicotin, and a part 
which is only slightly affected; the former being concerned with 
the more complicated movements, the latter with the simplest 
kind of reflex action. 

On account of the lasting action of nicotin on certain functions 
of the ganglia, i.e. probably on certain nerve-cells, it is not unlikely 
that a morphological change may be visible in some of the nerve- 
cells of each ganglion, if the ganglia be observed some time after 
nicotin has been given. But as this point requires a careful study 
of the normal structure of the ganglia of the crayfish, I reserve it 
for a later communication. 

Electrical stimulation of the nervous chain in a crayfish a short 
time after nicotin has been given to it differs in its effect from that 
produced in a normal crayfish; the contraction is less, and is limited 
to one or two segments above and below the point of stimula- 
tion ; apparently each ganglion sends fibres to two or three seg- 
ments, and the effects observed are due to a stimulation of these 
nerve-fibres after they have left the ganglion ; whereas the other 
ascending and descending nerve-fibres end in nerve-cells in the 
various ganglia; where in consequence of the action of nicotin, 
the nervous impulse set up in the fibres is stopped. 

(2) On a new species of Phymosoma. By Akthtjr E. Shipley, 
M.A., Christ's College. 

During a visit to the Bahama Islands, Mr Weldon was fortu- 
nate enough to find three specimens of a large brown Phymosoma, 
whilst investigating the Fauna of the Bimini lagoon. He came to 
the conclusion that these specimens belonged to no described 
species of Phymosoma, and was good enough to hand them over 
to me for description. I propose to call this species Phymosoma 
Weldonii. 

The length of the three specimens varied between 3'5 cm. 
and 3 cm. ; their bodies are plump and slightly curved. The 
ground colour of the preserved specimens is light yellow, but this 
is modified over the surface of the body by dark brown papillae. 
In all three specimens the introvert is retracted, and in this condi- 
tion is about 1 cm. long. The papillae are of two kinds, flat, brown, 
rectangular, low elevations on the skin of the trunk, and conical, 
elevated protuberances of a light colour on the introvert. 



78 Mr J. G. Adami, On the action of the [May 12, 

No hooks or traces of hooks were found on the introvert. 

At the base of the introvert, j ust behind the head, is a well- 
developed collar, such as I have described in detail in Phymosoma 
varians. 

The mouth is surrounded by a vascular lip, which at the dorsal 
middle line is continuous with the base of the lophophor. The 
latter is in the shape of a double horseshoe, and is composed of 
from 70 to 80 tentacles. 

There is nothing to call for remark in the arrangement of the 
internal organs, with the exception of the fact that there are only 
two retractor muscles. Such an arrangement is only met with 
elsewhere in Ph. Ruppellii from the Red Sea. The absence of 
hooks and of any traces of them is striking, but it occurs in five 
other species out of a total of 28 described. 

Habitat ; the Bahama Islands, Bimini lagoon. 

(3) On the action of the Papillaj^y Muscles of the Heart*. By 
J. George Adami, M.A., M.B., Christ's College. 

From time to time during the last twenty years continental 
observers have suggested that, in order to explain certain clinical 
phenomena, the papillary muscles of the ventricles must be looked 
upon as either contracting in a different manner to, or later than the 
rest of the heart, and in this country Ringer has suggested that 
one form of irregularity of the heart's action is due to a want of 
synchronism between the contractions of the muscles in question 
and the ventricular wall. On a priori grounds it would seem most 
unlikely that the papillary muscles contracted absolutely synchro- 
nously with the ventricular wall, for, were this the case, they 
would apparently nullify themselves. Attached as they are by the 
chordse tendinese passing from their apices to the edges and under- 
surface of the flaps of the auriculo-ventricular valves their main 
use is to aid in the complete closure of these valves, and thus to 
prevent regurgitation of blood into the auricles when the ventri- 
cles contract. And such is their position with relation to the 
auriculo-ventricular orifices that did they rapidly shorten contem- 
poraneously with the sharp beginning of the general ventricular 
contraction, at a time, this is, when the blood pressure in the 
ventricular chamber has not been greatly raised, in the absence of 
such sufficient counteracting pressure upon the under surface of 
the valves they would the rather pull the flaps of the valves apart, 
and aid regurgitation. Yet up to the present no one has to my 

* This paper embodies the results of a research that Professor Roy and I have 
been engaged upon conjointly during the last year. Fuller details will be found 
in a series of articles upon the 'Heart-beat and Pulse-wave,' published in the 
Practitioner. 



1890.] Papillary Muscles of the Heart 79 

knowledge endeavoured to obtain a simultaneous record of the 
work performed by these two portions of the cardiac muscle — nor 
has the probability that the two have different periods of action 
become in any way a part of general medical doctrine. Doubtless 
physiologists and medical men have voluntarily neglected to 
attempt exact observations upon this subject, from their knowledge 
that it is almost impossible satisfactorily to solve what would seem 
to be a much more simple matter, — the relation in time of the 
contraction of the muscle of the base and of the apex of the ven- 
tricle. And thus it has come to pass that nothing has been 
accomplished in what Professor Roy and I now find to be a field 
for research yielding suggestive, if not rich and important results. 

Without the aid of diagrams it would be difficult and tedious 
to describe fully the apparatus employed by us in our work upon 
this subject, but some idea may be given of the mechanism of our 
instrument. In order to understand as fully as possible the nature 
of the papillary contraction it is necessary to gain at the same 
time a tracing of the contraction of the ventricular wall, so that 
the relation in time of the different portions of either curve 
obtained may be interpreted in terms of the other. We found 
that the most satisfactory means of recording the ventricular wall 
contraction was as follows. Taking two points upon the anterior sur- 
face of the left ventricle, one near to the apex, the other nearer to 
the base, to the former was attached a light but firm rod, moving 
easily upon an axis so as not to prevent or modify the general to- 
and-fro movement of the heart ; into the latter was inserted a hook 
having a long shank, and these two parts of the instrument were 
connected in such a way with each other, and by means of a fine 
thread with the recording lever, that approximation of the two 
points upon the cardiac surface, that is to say, contraction of the 
muscle of the ventricular surface, resulted in an upward movement 
of the lever point ; separation and cardiac dilatation, in a down- 
ward movement. 

If now the light rod forming one limb of the above arrange- 
ment were attached to the surface of the ventricle over the 
region of insertion of one of the papillary muscles, a record 
of the contraction of this muscle could be simultaneously ob- 
tained by fixing on to the rod a cross-bar having at its further 
extremity a pulley, round which passed a fine thread, attached at 
one end to a second lever, and at the other to a strong hooked 
wire pulling upon one of the flaps of the mitral valve, and by this 
means upon the papillary muscle. To gain this attachment it was 
necessary to clamp off temporarily the left auricular appendix 
from the rest of the auricle, to make a small incision through its 
walls, to pass into this incision the hooked wire, to ligature the 
collar, in which the wire worked, to the wall of the appendix, and 



80 Mr J. G. Adami, On the action of the [May 12, 

then, removing the clamp, to pass the wire down through the 
mitral orifice and hook it over the edge of the mid-portion of one 
of the valve flaps. This operation required a certain amount of 
practice; the form of the curve shewed when it had been rightly 
performed. Attached in this way the hook pulled upon the free 
edge of the valve, and so through the chordae tendineaj upon the 
papillary muscles, while the wire moved freely to and fro through 
the collar inserted in the auricular wall ; there was little or no 
disturbance by clotting. The two levers recorded simultaneously 
upon the revolving drum, the one the contraction of the ventricu- 
lar wall, the other the approximation and separation of the base of 
the columna carnea and the edge of the mitral valve, that is to 
say, the contraction and expansion of the papillary muscle. 

Tracings so obtained shewed that the papillary muscles begin 
to contract and pull upon the mitral valves at a very definite 
interval after the commencement of the ventricular systole, indeed 
the interval between the two is so well marked that the papillary 
curves frequently exhibit an initial depression, due it would seem 
to an actual stretching of the papillary muscles and separation of 
their bases from the edges of the valve, the increased blood 
pressure within the ventricular chamber acting upon the valve 
and tending to drive it upwards into the auricular cavity. This is 
followed by the rapid contraction of the papillary muscles, which 
in its turn affects the curve obtained from the ventricular wall. 
Up to the moment when the papillary contraction begins, the lever 
point registering the heart-wall curve had ascended rapidly and 
in an almost straight line ; but now the ascent is slowed, and at 
times there may be a slight depression or actual notch on the 
upstroke. This does not indicate that there has been an inter- 
ference with the act of contraction on the part of the heart- wall, 
but that the shortening of the contracting muscles has been inter- 
rupted, and this lessened shortening is due to the sudden increase 
in the intra- ventricular blood pressure consequent upon the papil- 
lary contraction. Small in bulk as they are compared with the 
heart- wall, the musculi papillares by pulling upon the large flaps 
of the mitral valve must exert an influence upon a very consider- 
able proportion of the surface of the ventricular cavity, and their 
contraction must have a distinct effect upon the intra-ventricular 
blood pressure. That this is the case is rendered evident by a 
comparison of the heart-wall curve with the curve of intra-ventri- 
cular pressure. The slowing or depression upon the upstroke of 
the former corresponds in time with a very well-marked rise or 
secondary wave upon the latter, and it is this sharp rise of pres- 
sure due to the contraction of the papillary muscles that hinders 
for the time the shortening of the muscle fibres of the wall of the 
ventricle : these fibres suddenly receive, as it were, an extra load, 



1890.] Papillary Muscles of the Heart. 81 

and though there is no stoppage in the act of contraction the rate 
and extent of their contraction are in consequence diminished. 

The sudden powerful contraction of the musculi papillares is 
followed by a stage in which the shortening of the muscles is 
slowed and the ascent of the curve more gradual : and simul- 
taneously there is a more rapid ascent of the heart-wall curve. 
After this both portions of the ventricle remain for a comparatively 
long period in a state of contraction unaccompanied by further 
shortening, and the summits of both tracings are more or less 
flattened. We then find that the papillary muscles begin to 
expand before the rest of the ventricle. 

To sum up the above details : the papillary muscles begin to 
contract later than the ventricular walls, and commence their 
expansion at an earlier period. They act indeed only during that 
period when upon a priori grounds we shoidd expect them to be 
contracted, not pulling upon the segments of valve until these 
have been brought into firm apposition by the increased blood 
pressure, beginning to act also at a time when further increase 
of pressure would tend to drive the segments upwards into the 
auricle, and so cause regurgitation. Their contraction produces 
a sudden definite increase in the intraventricular blood pressure, 
well marked upon the blood pressure curves, and this increase 
causes a diminution in the rate of shortening of the muscle of 
the heart-wall, indicated by a depression upon the line of ascent 
of the curve obtained from the ventricular wall. 

We hesitate to offer any explanation of this virtually inde- 
pendent action of the papillary muscles : we can only declare that 
the more we have studied the tracings obtained under various 
conditions, the more we have been led to conclude that the 
moment when they begin to contract is not primarily dependent 
upon the moment of commencing ventricular contraction. We 
find for example that an overdose of liquor strychnine may lead 
to complete a synchronism between these two components of the 
ventricular action ; or, again, there may be a ventricular systole 
unaccompanied by papillary contraction, or vice versa. Again, the 
first effect of strophanthus is to cause rapidly increasing force of 
the contraction of the papillary muscles as compared with the 
heart-wall. Further, the period of papillary contraction bears no 
direct relation to the moment of origin of the pulse wave, to the 
time that is when the blood begins to pour from the heart 
into the arteries. Yet under normal conditions the pulse wave 
would seem to begin almost at the moment when the musculi 
papillares exert their first sharp strong pull upon the valves, and 
so act as an additional factor in raising the intraventricular 
pressure above that in the large arteries. In short, the phenomena 
of the papillary contraction would appear to supply further proof 



82 Mr J. 0. Adami, On the action of the [May 12, 

as to the automatic, non-nervous action of cardiac muscle-fibres. 
No nerve ganglia, and, as far as we know, no nerve fibres have 
been made out as controlling the musculi papillares, and the 
moment at which they begin to contract, the duration, and the 
extent of their contraction would appear to be determined in large 
measure by the intraventricular blood pressure and the quality of 
the blood. 

In conclusion, a few words may be said as to the way in which 
our observations throw light upon certain peculiarities of the 
pulse curve, which so far have been very variously explained — 
and as to which there has been much uncertainty. In tracings of 
the normal pulse gained by Marey's sphygmograph, or the equally 
unsatisfactory modification thereof usually employed in this 
country, or again by Dudgeon's sphygmograph in what may be 
termed its lucid intervals, there can often be seen two well- 
marked secondary waves in the first part of the curve previous 
to the dicrotic notch. The first of these has received the name of 
'apex' or 'percussion' wave, the second that of 'tidal' or 'pre- 
dicrotic.' That the former is not simply due to inertia is shewn 
by the fact that not unfrequently a small inertia wave may be 
superposed upon it, removable by proper adjustment of the instru- 
ment. By comparing the curves of the contraction of the ven- 
tricular wall or of the intraventricular blood pressure with the 
pulse curve taken simultaneously at the base of the aorta, Professor 
Roy and I have been enabled to shew that the first of these curves 
corresponds in time to the first period of contraction of the papillary 
muscles and the consequent increase in the intra-cardiac (and 
intra-arterial) blood pressure. This should therefore be termed 
the papillary wave. The second we consider is not by any means 
a secondary wave, but is really the latter portion of main wave due 
to the general ventricular systole, the first smaller papillary wave 
being superposed upon its first portion. This we would call the 
systole remainder ivave, or, more shortly, remainder wave. 

The same series of observations has also given us an explana- 
tion of the form of pulse usually termed the anacrotic, in which 
there is a small well-marked wave upon the upstroke of the pulse 
tracing, not, as in normal conditions, forming tbe apex of the curve. 
This form is to be found in cases where there is high intra-arterial 
pressure, or obstruction to the onward flow of the blood. Where 
there is high intra-arterial pressure there also the intra-cardiac 
pressure must be raised to a correspondingly high point before it 
becomes greater than that in the aorta, and before the valves be 
thrown open, that is to say, the pulse wave must begin at a later 
period of the cardiac systole. I have already stated that there is 
no absolute relation between commencement of the papillary 
contraction and the moment of opening of the aortic valves. 



1890.] Papillary Muscles of the Heart. 83 

Hence in this case a fair portion of the papillary contraction has 
taken place before the blood begins to pass from the ventricle, or 
to speak more correctly, before the pulse wave can be propelled 
along the aorta, consequently the papillary contraction is shewn 
but incompletely upon the pulse wave, only its latter part is 
represented in the pulse, the papillary wave appears at a lower 
point on the ascent of the curve than under normal conditions, 
the greater portion of the blood being expelled by the long con- 
tinuing systolic contraction; in fact, the jxipillary factor of the 
pulse is small, the systolic remainder considerable. It is interesting 
to note that where the intra- arterial pressure is greatly increased 
the intra-cardiac pressure curve shows the same tendency toward 
anacrotism ; the papillary wave, instead of forming the apex of the 
curve, may be comparatively low down upon the line of ascent. 



(4) Mr S. F. Harmer exhibited some living specimens of a 
Land-Planarian (Rhynchodemus terrestris, 0. F. Miiller) found in 
Cambridge. This animal was first described as a native of England 
by Rev. L. Jenyns (Observations in Natural History, London 1846), 
who discovered it in abundance in the woods of Bottisham Hall, 
near Cambridge. In the present instance, a search (made by 
kind permission of R. B. Jenyns, Esq.) in the same locality resulted 
in the discovery of a few specimens; and it was ascertained subse- 
quently that R. terrestris is by no means uncommon in Cambridge 
(King's College, Botanic Gardens). It may readily be found 
by examining the damp lower surface of logs of wood which have 
been lying for some time on the ground. Since the first discovery 
of the animal in England, it seems to have been very seldom 
found : but from its wide distribution in Europe generally and 
in England, and from the fact that it is not very likely to be 
found unless it is specially looked for, it is probable that this 
animal is much commoner than is usually supposed. Several egg- 
capsules of R. terrestris were discovered on May 15, on examining 
fragments of rotten wood among which some specimens of the 
animal had been kept for a week. 



VOL. VII. PT. II. 



84 Prof. Liveing, On Solution [May 26, 

May 26, 1890. 

Mr J. W. Clark, President, in the Chair. 

The following Communications were made to the Society : 

(1) On Solution and Crystallization. III. Rhombohedral and 
Hexagonal Crystals. By Prof. Liveing. 

(Abstract.) 

In a former communication (Trans. Vol. XIV.) the author had 
suggested, in order to account for hexagonal crystals, an arrange- 
ment of molecules defined by supposing space to be divided by 
planes into right triangular prisms, and a molecule placed at each 
corner of the prisms. No mechanical reason, however, was forth- 
coming to account for the molecules assuming such an arrange- 
ment. On the other hand if we suppose the excursions of the 
parts of the molecule to be comprised within an ellipsoid of definite 
dimensions for each kind of matter, and suppose that in passing 
into the crystalline state these ellipsoids pack themselves as 
closely as possible (which they will do if they attract each other 
according to any law), the surfaces of minimum tension will be 
certain planes having the symmetry observed in crystals. That 
symmetry depends on the dimensions of the ellipsoids. If they be 
spheroids, and if oblate have their axes in any ratio except 2:1, 
and if the orientation of the axes be such that each spheroid is 
touched by six others in points lying in a plane perpendicular to 
the axis of symmetry, the arrangement will be the same as if space 
were divided into equal rhombohedra and a molecule placed at 
every corner. In this case the surfaces of minimum tension will 
be planes having a rhombohedral symmetry. But in arranging 
the spheroids so as to place the greatest number in a given space 
there are two arrangements with the centres in planes perpendi- 
cular to the axis of symmetry which give the same number of 
spheroids per unit of volume, and are therefore so far equally 
probable. These two arrangements correspond to twin crystals 
when the twin axis is the axis of symmetry. A crystal formed of 
such molecules may therefore, so far as the packing of the mole- 
cules determines its structure, be built up of alternate layers, of 
no particular thickness, of twin crystals. If the external form be 
rhombohedral such alternations will in general give ridged faces, 
for which the surface tension will not be the minimum. But if 
the external form be any one (or more) of those known as hexago- 
nal, it will be identical for the two individuals of the twin, and 
this circumstance will determine the growth of hexagonal forms in 
cases in which the surface tension of the hexagonal forms is not 
much greater than that of rhombohedral forms. At the same time 



1890.] and Crystallization. So 

the fact that hexagonal forms lend themselves to the production of 
approximately globular masses with a minimum total surface, will 
increase the tendency to the development of hexagonal forms. 
We may still however get rhombohedral forms in some cases. It 
has been shown that if the molecular ellipsoids be oblate sphe- 
roids with their principal diameters in ratio of V2 : 1, and they be 
arranged so that each be touched by four others in the plane of its 
priucipal circular section and by four others in each of two other 
planes parallel to the first, the crystal will have cubical symmetry, 
and if we suppose the system uniformly strained in the direction 
of one of the diagonals of the cubes the spheroids will be deformed 
into ellipsoids and the crystal will have rhombohedral symmetry. 
In this case the alternations of twins will not be equally probable 
and the rhombohedral forms will generally predominate. Formula; 
are given for calculating the relative probability of different forms 
when the angular element of the crystal is known, and their appli- 
cation shown by examples. 



(2) On the Curvature of Prismatic Images, and on Amici's 
Prism Telescope. By J. Larmor, M.A., St John's College. 

It is well known that, in homogeneous light, a prism acts as a 
telescope in magnifying transversely the dimensions of objects in 
the field of view, while their longitudinal dimensions remain un- 
changed. In fact an incident parallel beam emerges as a parallel 
beam, so that the prism forms a telescopic system ; and the trans- 
verse magnification along any ray is, by the general law applicable 
to such systems, equal to the inverse ratio of the breadths of these 
incident and emergent pencils. 

These considerations apply equally to any battery of prisms. 

As a single prism has a position of minimum dispersion which 
is different from the position of minimum deviation, it follows that 
two prisms, of the same kind of glass, may be combined in opposing 
fashion so as to form an achromatic pair, while some deviation 
remains, and therefore also some magnification. By combining in 
perpendicular planes two such achromatic doublets, of equal magni- 
fying power, Amici long ago succeeded in producing a telescope 
which magnified equally (about 4 times) in all directions, was 
made of the same kind of glass throughout and yet achromatic, 
and, according to Sir John Herschel's experience of it (Encyc. 
Metrop. 'Light,' § 453), gave images of remarkable perfection. 

The image of a straight-edge or slit, seen through a prism with- 
out a collimator, is a curved arch or bow : and it has been pointed 
out by Sir Howard Grubb that when the prism is rotated the 
curvature of this arch is proportional to the dispersion of the 



86 Mr Larmor, On the Curvature of Prismatic [May 26, 

spectrum produced. This law will be formally established below 
for any cylindrical optical system whatever which is composed 
throughout of the same kind of glass : it may be readily verified by a 
cursory examination of a pair of prisms standing on a flat plate. It 
follows from it that each of Amici's doublets gives images of straight 
lines which are free from curvature for the very reason that they 
are free from chromatic defect ; and the remarkable absence of dis- 
tortion noticed by Sir John Herschel is explained. 

We proceed to obtain a formula for the curvature of the image 
of a vertical slit seen a distance a through a system of prisms and 
cylindrical lenses of the same material, standing on a horizontal 
plane. The horizontal projection of a ray which is travelling at an 
inclination 6 to the horizontal plane — and is therefore refracted to 
an inclination 0', given by the same law sin = fi sin & as that of 
Snell — will be refracted according to the variable index /u, cos & /cos 0, 
or approximately fi+ \ (fi — fT 1 ) &\ as is small. This principle, as 
was pointed out by Stokes, will suffice for the solution of the 
problem. 

Thus the coordinates of a point on the image are 

x= ad, 

dD 

where D is the deviation of a horizontal ray. 

The curvature is equal to 2y/x 2 , and is therefore 

fx — fxT 1 dD 

a dfi ' 

dD . 
wherein -r— is clearly the angular dispersion of the spectrum pro- 
duced by the combination. 

The investigation is no more difficult for a slit inclined at an 
angle e to the vertical. In this case 

x= a0, 

y = aO tan e + a \ -j-r 6 tan e 4- -j- \ (/a — /uT 1 ) 2 i ; 

therefore 

/, dD\ 1 dD , _ u 2 

^- taD H 1+ #r = 2a^ (/i ~^ )X - 
Thus the image is parabolic, of the same curvature 

p — fuT 1 dD 
a dfi ' 



1890.] Images, and on Amici's Prism Telescope. 87 

as above ; and 

tan 7] _ cW 
tan e dcf> ' 

where t] is the inclination of the image to the vertical, and <f> is the 
angle of incidence of the axial ray on the first face. 



(3) On some theorems connected with Bicircular Quartics. 
By R. Lachlan, M.A., Trinity College. 

The object of this paper is to extend to Bicircular Quartics, 
and twisted quartics, Sylvester's theory of residuation in connection 
with the plane cubic. 

1. A curve of the 2nih order, having multiple points of the 
nth order at each of the circular points, may be called a circular 
curve of the 2?ith order. Such a curve is determined by n (n + 2) 
points, as is seen at once by writing down its equation ; and two 
circular curves of the 2nth and 2mth orders intersect in 2mn points, 
other than the circular points. 

Let U, V be any two circular curves of the 2nth order passing 
through n(n + 2) — l given points, then U+kV will represent 
any circular curve of the 2?ith order passing through these points, 
bat such a curve must obviously pass through all the points in 
which U and V intersect. Hence it follows that any circular 
curve of the 2nih order which passes through n (?z + 2) - 1 fixed 
points must pass through (n — l) 2 other fixed points. 

2. Further we may show that every circular curve of the 2nth 
order which passes through 2np — (p — l) 2 points on a circular 
curve of the 2pth order (p being < n) meets this curve in (p — l) 2 
other fixed points. For if we draw any circular curve of the 
2(n— p)th order through (n — p) (>i—p + 2) assumed points, then 
since 

2np -(p- l) 2 + (n -p) (n -p + 2) = n (n + 2) - 1, 

any circular curve of the 2nth order which passes through the 
given 2np — (p — If points on the curve of the 2pth. order and 
also through the assumed points on curve of the 2 (ii—p)th. order, 
must pass through {n — l) 2 other fixed point. But the given curve 
of the 2pth order and the curve of the 2 (n — p)th order make up 
one such circular system ; hence these (n — Yf points must lie on 
one or other of these curves of lower order. The most that can 
lie on the curve of the 2 (?i — jj)th order is 

2« (n - p) - (n - p) (n-p + 2), i.e. (n - 1/ - (p - l) 2 ; 



88 Mr Lachlan, On some theorems [May 26, 

hence the remaining (p — l) 2 points must lie on the curve of the 
2pth order. Hence the truth of the theorem enunciated is 
manifest. 

3. An important particular case may be thus stated : Every 
circular curve of the 2nth order which passes through 4?i — 1 fixed 
points on a bicircular quartic must pass through one other fixed 
point*. r 

4. From this theorem we may deduce the following : If of 
the 4<(m + n) intersections of a circular curve of the 2(m + ?i)th 
order with a bicircular quartic, 4>m lie on a circular curve of the 
2mth order, the remaining 4m lie on a curve of the 2/ith order. 

For let U m denote the curve of the 2mth order, and let a curve 
U of the 2?ith order be described passing through 4m — 1 of 
the remaining 4n points ; then these curves U m , U n together make 
up a circular system of the 2(m + ?i)th order passing through 
4 (m + n) — 1 points on a bicircular quartic, this curve must there- 
fore pass through one other fixed point, which must clearly lie on 
TJ n ; also this point must be a point in which the given curve of 
the 2 (m + ?i)th order meets the bicircular quartic ; hence we see 
that the theorem stated above must be true. 

5. If now we have two systems of points a, ft, which together 
make up the complete intersection of a bicircular quartic with 
a circular curve of any degree, i.e. if a + ft is a multiple of 4, one 
of these systems may be called the residual of the other. Since 
through a given system of points, any number of curves of different 
orders may be described, it is evident that a given system of points 
a has an infinite number of residual systems ft, ft', &c. Two 
systems of points ft, ft', may be called coresidual systems if both 
are residuals of the same system a. 

6. We have at once the following theorems : 

i. Two points which are coresidual must coincide. This is 
merely a restatement of the theorem in § 3 ; for if through 4n — 1 
points a on a bicircular quartic we describe two circular curves 
of the 2 nth order, meeting the quartic again in the points ft, ft', 
then ft and ft' are one and the same point. 

ii. If two systems ft, ft' be coresidual, any system a' which 
is a residual of one will be a residual of the other. 

Suppose that through any system a, two curves U p , U q are 
described meeting the quartic again in systems ft, ft', then by 
definition ft, ft' are coresidual systems ; then if through ft' a curve 

* Some interesting results connected with bicircular quartics, obtained by 
developing this theorem, were given in a paper communicated to the London 
Mathematical Society in May, 1890. 



1890.] connected with Bicircular Quarties. 89 

U r be drawn meeting the quartic again in the system of points a', 
then the systems ft, a' will also be residual. For since the systems 
a, ft make up the intersection of a curve U p with the quartic, and 
a', ft' make up its intersection with a curve U r , the four systems 
together make up the intersection with the quartic of a curve 
whose order is 2 (p + r) ; but the systems a, ft' together make up 
the intersection of the quartic with the curve U q of order 2q, and 
therefore by § 4, the systems a, ft together make up the complete 
intersection of the quartic with a curve whose order is 2 (p + r — q). 

iii. Two systems which are coresidual to the same are co- 
residual to each other. 

If ft and ft' are coresidual as having a common residual a, and 
if ft', ft" have a common residual a ; then by the last theorem a 
is also a residual of ft", and a' a residual of ft ; that is, if ft, ft" are 
each of them coresidual with ft', then ft, ft" are coresidual with 
each other, for a, a' are each of them a residual of ft, ft". 

7. Suppose now that we have given a system of 4<p + 1 points 
on a bicircular quartic, through them we may draw a circular 
curve of order 2 (p + r) and we obtain a residual system of 4r — 1 
points ; through these we may draw a circular curve of order 
2 (r + s) and the residual system will consist of 4s + 1 points ; 
through these we may draw a circular curve of order 2 (s + 1) and 
we obtain a residual consisting of 4£ — 1 points. If at auy stage 
where we have a residual of 4?i — 1 points we draw a circular curve 
through them of order 2?i we obtain a residual of a single point, 
and it follows from the theorems stated in § 6 that this point 
must be the same whatever be the process of residuation. More- 
over whatever system of points we start with, either a system of 
4^> + 1 points or a system of 4p — 1 points, we can always by an 
even or odd number of stages, obtain a single point which will be 
a coresidual or a residual of the given system, according as the 
number of points in the given system is 4>p + 1 or 4jj — 1. 

The principles just established enable us to find, by means of 
circular constructions, the point residual or coresidual to any given 
system of points, the number of which is 4p ± 1. 

8. To find the coresidual point of a system of five given points. 
Let P v P 2 , P 3 , P 4 , P 5 be the points, through any three of these 

Pj, P 2 , P 3 say, draw a circle cutting the quartic in Q v and through 
the other two P 4 , P 5 draw a circle cutting the quartic in Q 2 , Q s , 
then the circle QiQ»Q 3 will cut the quartic in the point R which 
will be the coresidual of the given system. 

Let the points P l5 P 2 , P 3 , P 4 , P 5 coincide, then we see that to 
obtain the coresidual of five consecutive points P, we have to 
draw the circle of curvature at P meeting the quartic in Q, and 
one of the bitangent circles at P touching the quartic again at Q x , 



90 Mr Lachlan, On some theorems [May 26, 

then the circle which touches the quartic at Qj and passes through 
Q will cut the curve again in P the coresidual of five consecutive 
points at P. Hence we have the theorem that if the bitangent 
circles at P touch the bicircular quartic again at the points 
Qj, Q 2 , Q 3 , Q 4 and the osculating circle at P cut the curve again 
in the point Q, then the four circles which can be drawn passing 
through Q and touching the quartic at Q v Q 2 , Q 3 , Q 4 cut the 
quartic again in the same point P. 

Again if we draw a bicircular quartic passing through the five 
points Pj, P 2 , P 3 , P 4 , P s it must cut the given quartic in three 
points p % , p 2 , p 3 , the circle through which must pass through P 
the coresidual of the five given points ; hence if we wish to draw 
a bicircular quartic passing through five given points on a given 
bicircular quartic and osculating the latter elsewhere, we have 
merely to draw a circle osculating the given curve and passing 
through P the coresidual of the five given points ; but nine such 
circles can be drawn ; hence nine systems of bicircular quartics 
can be drawn passing through five given points on a given 
bicircular quartic, which have three-point contact with it elsewhere. 

9. To find the residual point of a system of seven given points. 
Let the points be P v P 2 ,...P 7 ; through any three of them such 

as P i; P 2 , P 3 draw a circle cutting the quartic in Q 1} through P 4 , P 5 
draw a circle cutting the quartic in Q 2 , Q 3 , and through P 6 , P 7 
a circle cutting the quartic in Q 4 , Q 5 . And then we may find P 
the coresidual of the system Q v Q 2 , Q 3 , Q 4 , Q 5 as in § 8, P will be 
the residual of the given system of seven points. 

Or we might replace any six of the points by their coresidual 
points; thus let the circle P l P 2 P 3 cut the quartic in Q 1} and the 
circle P i P s P a cut the quartic in Q 2 ; and then let any circle be 
drawn through Q t , Q 2 to cut the quartic in P^,P 2 ; which two 
points constitute a coresidual system of the system P t , P 2 ...P 6 . 
Then if the circle P/, P 2 ', P 7 cut the quartic in R, R will be the 
residual of the given system. 

By this method we are enabled to find the eighth point in 
which any bicircular quartic which passes through the seven given 
points cuts the given quartic, for every bicircular quartic through 
the seven given points must pass through P. This method 
assumes that one quartic through the seven points is given ; and 
thus the problem is not the same as finding the eighth point when 
only seven points are given. 

10. To find the residual point of seven consecutive points. 
Let the points P V ..P 7 in §9 coincide with the point P, let the 

circle of curvature at P meet the curve in Q, also let Q 1} Q 2 , Q 3 , Q 4 
be the points of contact of the bitangent circles at Q; then two 
consecutive points at Q l may be considered as a coresidual system 



1890.] connected with Bicircular Quartics. 91 

of the six points P t , P 2 ,...P 6 ; and then the circle touching the 
curve at Q 1 and passing through P v i.e. P, must meet the curve 
again in R the residual point of the seven consecutive points P. 

Otherwise, we may obtain R' the coresidual point of five 
consecutive points P as in § 8, and then the circle which touches the 
curve at P and passes through R' must meet the curve again in R. 

Incidentally we may notice that we have the theorem : if the 
circle of curvature at any point P of a bicircular quartic meet the 
curve again in Q, and if Q iy Q 2 , Q 3 , Q 4 be the four points of contact 
of the bitangent circles at Q, then the four circles which can be 
drawn touching the curve at these points respectively and passing 
through P, cut the curve again in the same point R. 

Hence we see that R can only coincide with P, when Q 1 ,Q 2 ,Q 3 , Q 4 
are the points of contact of the bitangent circles at P, in which 
case Q must coincide with P, i.e. P must be a cyclic point. 

Thus at any point P on a bicircular quartic we can in general 
draw other bicircular quartics having seven point contact at P, 
and they will all cut the curve again in the point R. If however 
P be a cyclic point these bicircular quartics will have eight-point 
contact with the given quartic at P. 

11. All these theorems admit of translation so as to apply to 
twisted quartics, we have merely to substitute in the enunciation 
of any theorem the word plane for circle. In fact we have only 
to prove that if any surface of the nth. order passes through 4n — 1 
fixed points on the curve of intersection of two quadrics, it must 
also pass through one other fixed point. 

Let U 2 , V 2 denote the two given quadrics, and U n any surface 
of the nth degree passing through 4n — 1 fixed points on the curve 
of intersection of U 2 , V 2 ; then we have to show that any other 
surface of the nth. degree passing through these fixed points will cut 
the curve of intersection of U 2 , V 2 in the same point as U n . Let 
a surface U n _ 2 of the (n — 2)th order be drawn passing through 
n (n — 2) arbitrary points on the curve of intersection of V 2 Avith 
U n , and also through ^{n — l)(n — 2)(n — 3) arbitrary points on 
the surface U n ; also let a surface V n _ 2 of the (n — 2)th order be 
drawn passing through n (n — 2) arbitrary points on the curve of 
intersection of U 2 with U n , and also through the same points on 
U n as JJ„ ,. Then we have three surfaces U n , U .U ,V , . V, 
each of the nth order and each passing through the same points, 
Avhose number is 

4ra - 1 + 2n (n- 2) + %(n -l)(n- 2) (n - 3) = ±n (n 2 + Qn + 11) - i>, 
which is two less than the number necessary to determine a surface 
of the nth degree, any surface of the nth degree therefore which 
passes through these points must be of the form 

u n + xu n _ 2 .u 2 + f ,v n _ 2 .v 2> 

VOI,. VII. PT. II. 8 



92 Mr Lachlan, On Bicircular Quartics. [May 26. 

and consequently must pass through all the points in which the 
three surfaces U n , U n _JI 2l V n ^V 2 intersect. Hence any surface 
of the nth degree which passes through the 4n — 1 points of 
intersection of the surfaces U. 2 , V 2 , U n must pass through the 
remaining point of intersection. 

12. Exactly as in § 4 we may deduce the theorem : if of the 
4 (m + n) intersections of a surface of the (m + n)ih order with the 
curve of intersection of two quadrics, 4m lie on a surface of the 
mth degree, then the remaining 4<n must lie on a surface of the 
nth degree. 

Also the definitions of residual and coresidual systems of points 
require such slight modification that it is needless to recite them. 
And in applying the principles of residuation we see that we have 
merely to substitute planes for circles, and thus we obtain what 
might be called ' planar ' constructions for finding the residual or 
coresidual point of any system of points on a twisted quartic. 

June 2, 1890. 

At a meeting of the Council of the Society, it was decided, in 
accordance with the Reports of the adjudicators, Sir W. Thomson, 
Lord Rayleigh, and Prof. G. H. Darwin, to award the Hopkins 
Prize for the period 1883—5 to W. M. Hicks, M.A., F.R.S., for 
his memoir upon the Theory of Vortex Rings (Phil. Trans. 1885) 
and for his earlier memoirs upon related subjects — also to award 
the Hopkins Prize for the period 1886—8 to Horace Lamb, 
M.A., F.R.S., for his paper on Ellipsoidal Current-Sheets (Phil. 
Trans. 1887) and for his numerous other papers on Mathematical 
Physics. 



PROCEEDINGS 



OF THE 



Cmkifr0£ IJjriksopjwal Batuty. 



October 27, 1890. 

ANNUAL GENERAL MEETING. 
Mr J. W. Clark, President, in the Chair. 

The following Fellows were elected Officers and new Members 
of Council for the ensuing year : 

President : 
Prof. G. H. Darwin. 

Vice-Presidents : 
Mr J. W. Clark, Prof. Babington, Prof. Liveing. 

Treasurer : 
Mr R. T. Glazebrook. 

Secretaries : 
Mr J. Larmor, Mr S. F. Harmer, Mr E. W. Hobson. 

New Members of Council: 
Dr A. Hill, Dr A. S. Lea, Mr A. Harker, Mr L. R. Wilberforce. 

The names of the Benefactors of the Society were recited by 
the Secretary. 

Fourteen names were proposed, with the approval of the 
Council, for election as Honorary Members of the Society. 

The retiring President, Mr J. W. Clark, before vacating the 
chair, gave a short sketch of the origin and early years of the 

VOL. VII. PT. III. 9 



94 Mr C. Chree, On some Compound Vibrating Systems. [Oct. 27, 

Society, which will be published as a separate part of the 
Proceedings. 

The President elect, Prof. G. H. Darwin, then took the Chair, 
and the following Communication was made to the Society : 

(1) On some Compound Vibrating Systems. By C. Chree, M.A., 

King's College. 

(Abstract.) 

The vibrating systems treated in this memoir are bounded 
either by concentric spherical or by coaxial cylindrical surfaces, 
and the vibrations are of those types in which the displacements 
are either wholly radial or wholly transverse. 

By a simple system is meant a spherical or a cylindrical shell 
of a single isotropic medium ; by a compound system is meant 
a stratified medium in which the surfaces separating adjacent 
media, or layers, are spherical or cylindrical according as the 
outer surfaces are spherical or cylindrical. 

Those functions which when equated to zero constitute the 
frequency equations for a simple system are termed frequency 
functions. A. method is developed whereby the frequency equation 
for a compound system of any number of layers, composed of 
different isotropic media, can be at once written down in a form 
which involves the frequency functions of the several layers. 

The general result so obtained is employed in determining the 
change in the pitch of the several notes in an otherwise isotropic 
simple shell owing to the existence of a thin intercalated layer 
of a different isotropic medium. The dependence of the magnitude 
of the change of pitch on the nature of the difference between 
this altered layer and the remainder of the shell, on the position 
of the altered layer, and on the value of Poisson's ratio for the 
unaltered medium is considered for the system of notes which 
the vibrating system in question is capable of producing. 

The law of variation of the magnitude of the change of pitch 
in solid spheres or cylinders with the position of an altered layer, 
which differs from the remainder of the system in a given assigned 
way, is represented by a curve or curves. Every such curve shows 
in a very simple manner the comparative magnitude of the largest 
possible changes of pitch, for all possible notes of the system, which 
can arise from a given alteration of material, throughout* a layer of 
given volume or of given thickness as the case may be. The cor- 
responding positions of the layer may also be immediately derived 
from the curves for all those notes of the system whose frequencies 
are recorded. 

Tables are constructed showing the positions where a thin layer 
differing in an assigned way from the remainder is most effective 



1890.] Dr Gamgee, On Fahrenheit's Thermometrical Scale. 95 

iu altering the pitch of several of the notes of lowest pitch, and 
other tables show the numerical magnitude of the corresponding 
maximum changes of pitch. 

The changes in the types of vibration in solid spheres or 
cylinders due to the existence of thin intercalated layers of other 
media are determined by a different method. This method also 
leads to expressions for the changes of frequency in these systems 
which are identical with those obtained by the first method. 

The results obtained in the memoir are very numerous, and 
many of them seem of interest from a physical point of view. 
They show that general laws as to the effects of altering the 
stiffness or elasticity of vibrating systems unless carefully restricted 
may lead to very erroneous conclusions. 



November 10, 1890. 

Professor G. H. Darwin, President, in the chair. 

The following Communications were made to the Society : 

(1) Note on the principle upon which Fahrenheit constructed 
his Thermometrical Scale. By Arthur Gamgee, M.D., F.R.S., 
Emeritus Professor of Physiology in the Owens College (Victoria 
University). 

[Abstract; reprinted from the Cambridge University Reporter, Nov. 18, 1890.] 

The author commenced by drawing attention to the fact that, 
although the Fahrenheit thermometer has been so generally used 
in England, no accurate information was to be found in our text- 
books concerning the principles upon which its scale had originally 
been constructed. He referred, however, to a view advanced by 
Professor P. G. Tait in his elementary treatise on 'Heat,' and 
which had been accepted by many teachers, according to which 
Fahrenheit divided his scale between 32° and 212° into 180 de- 
grees, in imitation of the division of a semi-circle into 180 degrees 
of arc. This theory rested on the incorrect supposition that, before 
Fahrenheit's time, Newton had suggested, as a basis for a thermo- 
metric scale, the fixing of the freezing and boiling points of water, 
the space between these being divided into a number of equal 
degrees. The author pointed out that in his " Scala graduum 
caloris," Newton made no such suggestion as that attributed to him 
by Professor Tait, and prior to him by Professor Clerk Maxwell ; and, 
indeed, that Fahrenheit had settled the basis of his scale and had 
constructed a large number of thermometers which were used by 
scientific men throughout Europe, many years before the discovery 
by Amanton (which Fahrenheit confirmed and gave precision to) 

9—2 



96 Mr Brindley, On the size of certain animals [Nov. 10, 

of the fact that under a constant pressure the boiling point of 
water is constant. 

The author stated that the thermometers which were first 
constructed by Fahrenheit were sealed alcoholic thermometers, 
provided with a scale in which two points had been fixed. The 
zero of the scale, representing the lowest attainable temperature, 
was found by plunging the bulb of the thermometer in a mixture 
of ice and salt, whilst the higher of the two points was fixed by 
placing the thermometers under the arm-pit or inside the mouth 
of a healthy man. The interval between these two points was, in 
the first instance, divided into 24 divisions, each of which corre- 
sponded to supposed well characterized differences in temperature, 
and each being subdivided into four. In his later alcoholic and 
mercurial thermometers, the 24 principal divisions were suppressed 
in favour of a scale in which 96 degrees intervened between zero 
and the temperature of man ; in these later thermometers the 
32nd degree was fixed by plunging the bulb of the thermometer 
in melting ice. 

The author then pointed out that Fahrenheit was led to con- 
struct mercurial thermometers in order to be able to ascertain the 
boiling point of water; with this object the scale constructed, as 
has been stated, was continued upwards, in some cases so as to 
include 600 degrees. 

It was as the result by experiment alone, that the number 212 
was obtained as the temperature at which water boils, at the mean 
atmospheric pressure. 

The author in conclusion argued that Fahrenheit took as the 
basis of his thermometric scale the duodecimal scale which he was 
constantly in the habit of employing. 



(2) On Variations in the Floral Symmetry of certain Flowers 
having Irregular Corollas. By William Bateson, M.A., Fellow 
of St John's College, and Anna Bateson. 

(3) On the nature of the relation between the size of certain 
animals and the size and number of their sense-organs. By H. H. 
Brindley, B.A., St John's College. 

[Abstract; received November 29, 1890.] 

In speculation as to the evolution of various forms it is gene- 
rally held as a principle, that the conditions of the struggle for 
existence are such that variations in the direction of atrophy or 
diminution in bulk of a useless organ must necessarily be bene- 
ficial by reason of the saving of tissue and effort which is effected 



1890.] and the size and number of their sense-organs. 97 

by this reduction. It has been assumed by many that this benefit 
must be so marked as to lead to the Natural Selection of the 
individuals thus varying. This principle has been invoked especi- 
ally in the case of sense-organs, and, for example, it has been 
suggested that the blindness of cave-fauna may have come about 
by its operation. 

With the object of testing the truth of this assumption, it 
seemed desirable to obtain a knowledge of the normal variations 
in size and number of sense-organs occurring within the limits of 
a single species. The cases chosen were (1) The olfactory organ 
of Fishes (Eel, Loach, Pleuronectidce, &c), and (2) The eyes of 
Pecten opercularis. In the first case tables were given shewing 
that large individual fluctuations occur, but that on the whole the 
number of olfactory plates, increases with the size of the body. 
It was pointed out that the size of the eye in Fishes also increases 
with the size of the body. 

In the case of Pecten, however, though the size of the eyes 
increases with the diameter of the animal, yet in specimens having 
a diameter of 3 cm. — 6 cm. the number of the eyes is not thus 
related (cp. Patten), but varies in a most surprising and, as it 
were, uncontrolled manner. 

Statistics were given shewing that in individuals of the same 
size, the number of eyes may vary between 70 and 100, and that 
no uniformity is to be found. It was pointed out that these eyes 
are large and complicated organs, having lens, retina, tapetum, 
&c, involving great cost in their production. These facts suggest 
that the " economy of growth " cannot be a principle of such 
precise and rigid character as to warrant its employment as a basis 
for speculation as to the mode of evolution of a species. The 
diverse results in the case of the two sets of organs examined 
further indicate that the problem is one of far greater complexity 
and shews clearly that argument from analogy is inadmissible in 
these cases. 

(4) On the Oviposition of Agelena labyrinthica. By C. War- 
burton, B.A., Christ's College. 

[Abstract; reprinted from the Cambridge University Reporter, Nov. 18, 1890.] 

The oviposition and cocooning of Agelena labyrinthica is a 
striking case of the performance of a series of complicated opera- 
tions in obedience to a blind instinct. 

The eggs are always laid at night, but the presence of artificial 
light is quite disregarded by the animal. 

For about 24 hours before laying, the spider is engaged in 
preparing a chamber for the purpose. 



98 Mr Warburton, On spiders near Cambridge. [Nov. 10, 

Near its roof a small sheet is then formed, and the eggs are 
laid upwards against it and are covered with silk. A box is then 
constructed with this sheet as its roof, and is firmly attached by 
its angles to the roof and floor of the chamber. This box is con- 
structed and jealously guarded even if the eggs are removed imme- 
diately on oviposition. 

The whole operation involves about thirty-six hours of almost 
incessant industry. 

(5) Supplementary list of spiders taken in the neighbourhood 
of Cambridge. By 0. Warburton, B.A., Christ's College. 

[Received November 14, 1890.] 

In Vol. VI. of these Proceedings a list was given of some hun- 
dred species of local Aranese. To these must now be added the 
following, some of which have been taken since the former publi- 
cation, while others are inserted on the authority of the Rev. 
O. Pickard-Cambridge, who has kindly furnished a list of Spiders 
sent to him some years ago by the late Mr Farren. 

Unfortunately Mr Farren did not record the exact locality nor 
the frequency of his captures, but he is known to have carefully 
searched Wicken Fen, which is probably the habitat of most of his 
species. 

DYSDERIDAE. 
Dysdera 

crocota, C. L. Koch, rare, Castle Hill. 
Segesteia 

senoculata, Linn. 

DRASSIDAE. 

Dkassus 

troglodytes, C. L. Koch, rare, Wicken Fen. 
blackwallii, Thor. 

Clubiona 

corticalis, Walck., rare, University bathing enclosure. 

reclusa, Cambr. 
Anyphaena 

accentuata, Walck. 
Phktjkolithus 

festivus, C. L. Koch, Fleam Dyke. 

DICTYNIDAE. 

DlCTYNA 

latens, Fabr. 

AGELENIDAE. 
Hahnia 

nava, Bl., rare, Wicken Fen. 
Lethia 

humilis, Bl. 



1890.] Mr Warburton, On spiders near Cambridge. 99 

THERIDIIDAE. 

Thekidion 

simile, C. L. Koch. 

tiucturn, Walck., common, on shrubs and bushes. 

rufolineatum, Luc. 
Neeiene 

cornuta, BL, rare, in the "Backs." 

nigra, BL, rare, Turf Fen, Chatteris. 

fuscipalpis, C. L. Koch. In the bathing enclosure. 

apicata, Bl. 

bicolor, Bl. Castle Hill. 

bituberculata, Wid. 
Walckenaeea 

bifrons, Bl. 

unicornis, Bl. 

cristata, Bl., rare, Christ's Coll. Garden. 
Pachygnatha 

listeri, Sund. 
Eueyopis 

blackwallii, Cambr. 
Linyphia 

nigrina, Westr. 

setosa, Cambr., Wicken Fen. 

clathrata, Sund., Wicken Fen. 

circumspecta, Bl., rare. 

EPEIRIDAE. 
Epeiea 

acalypha, Walck. 
solers, Walck. 

quadrata, Clrk., occasional, Fens. 
Eeeata, 

In the previous list, Amaurobius fenestralis should have been recorded 
as rare instead of common, and the habitat of Theridion varians as 
being " boathouses and out-buildings" rather than "bushes". 

November 24, 1890. 
Prof. G. H. Darwin, President, in the Chair. 

The following gentlemen, duly nominated by the Council, were 
elected Honorary Members of the Society : 

Francesco Brioschi ; on the ground of his contributions to 
mathematical science by his investigations in the theory of forms, 
the theory of equations, and in elliptic and hyperelliptic functions. 

Leopold Kronecker ; on the ground of his contributions to 
mathematical science by his investigations in the theory of 
numbers and elliptic functions. 

Sophus Lie; on the ground of his contributions to mathe- 
matical science by his investigations in geometry, in the theory of 
differential equations, and in the theory of groups. 

Henri Poincare ; on the ground of his contributions to mathe- 
matical science by his investigations in the theory of functions 
and in mathematical physics. 



100 Honorary Members of the Society. [Nov. 24, 

George William Hill ; on the ground of his contributions to 
astronomical science by his investigations on the secular motion of 
the Moon's perigee and other researches in the lunar theory. 

J. Willard Gibbs ; on the ground of his contributions to 
physical science and specially to the sciences of thermodynamics 
and electromagnetism. 

Heinrich Hertz; on the ground of his contributions to 
the science of electromagnetism and specially for his brilliant 
experimental verification of Maxwell's theory. 

Arthur Schuster; on the ground of his contributions to 
physical science and specially for his researches on spectrum 
analysis and on the passage of the electric spark through high 
vacua. 

Victor Meyer ; on the ground of his contributions to chemical 
science, namely his researches on the nitro compounds of the fatty 
series, on the thiophenes, on pyro-chemistry, his development of 
Raoult's researches, and many other investigations. 

James D wight Dana ; on the ground of his contributions to 
mineralogical and geological science, namely his researches on 
coral islands, his great work A System of Mineralogy, and numerous 
other papers. 

Henry Bowman Brady; on the ground of his zoological re- 
searches and in recognition of his generosity in presenting to the 
University a valuable collection of Foraminifera. 

Rudolf Heidenhain; on the ground of his contributions to 
physiology, dealing with the physiology of secretion and absorption, 
and the physiology of muscles. 

Elias Metschnikoff ; on the ground of his researches in 
many fields of biological science, and especially in the study of 
embryology. 

Melchior Treub, Director of the Botanical Gardens, Java; 
on the ground of his general researches in botany. 

The following were elected Fellows of the Society : 
S. Ruhemann, M.A., Gonville and Gaius College. 
A. W. Flux, B.A., Fellow of St John's College. 
H. H. Brindley, B.A., St John's College. 

The following were elected Associates : 

David Sharp, M.B. (Edin.), F.R.S., Curator in Zoology. 

H. Gotobed. 

The following Associates were re-elected for a further period of 
three years : 

R. Bowes, R. I. Lynch, 

J. Carter, W. E. Pain, 

A. Deck, W. W. Smith. 



1890.] Mr Bryan, On a revolving cylinder or bell. 101 

The following Communications were made to the Society : 

(1) On the beats in the vibrations of a revolving cylinder or bell. 
By G. H. Bryan, M.A., St Peter's College. 

In this paper I propose to investigate the nature of the beats 
which may be heard when a vibrating shell in the form of a 
cylinder or other surface of revolution has imparted to it a rotatory 
motion about its axis of figure. 

It might at first appear that, unless such a body were revolving 
with angular velocity comparable with the frequencies of the 
vibrations, the latter would not be affected in any sensible manner, 
and that the only important effect of rotation would be in per- 
manently straining the body, owing to centrifugal force. This is, 
for example, the point of view taken by Mr Love in his paper on 
" The free and forced vibrations of an elastic spherical shell con- 
taining a given mass of liquid,"* in which the author uses the 
expressions for the accelerations referred to moving axes when 
dealing with the oscillations of the liquid, but not when dealing 
with those of the elastic envelope, although both are supposed to 
rotate together. But, while we may be justified in neglecting the 
effect on any single period of such small changes in the system as 
those due to rotation, yet the slight opposed changes produced in 
the periods of two similar vibration forms which travel in different 
directions round the shell may produce phenomena of beats, which 
in the case of very rapid vibrations like those of sound, are among 
the most noticeable effects of the rotation. 

If a straight wire of circular section, clamped symmetrically at 
one end, be made to rotate slowly about its axis while executing 
transverse vibrations, it is well known that the plane of vibration 
will remain fixed in space instead of turning with the wire. If the 
vibrations are audible we shall, therefore, hear a continuous sound. 
In the case of a tuning-fork the plane of vibration must necessarily 
turn with the fork, so that beats are heard if it be rotated. 

When however the vibrating body is such as a bell, rotation 
about its axis will produce an intermediate effect by causing the 
nodal meridians-{- to revolve with angular velocity less than that of 
the body, and depending in each case on the mode of vibration 
considered. This phenomenon, which forms the subject of the 
present paper, appears to be new, yet nothing is easier than to 
verify it experimentally. If we select a wine-glass which when 
struck gives, under ordinary circumstances, a pure and continuous 
tone, we shall on twisting it round hear beats, thus showing that 
the nodal meridians do not remain fixed in space. And if the 
observer will turn himself rapidly round, holding the vibrating 

* Proc. Lond. Math. Soc. xix. 

t That is meridians along which the vibration has no radial component. 



102 



Mr Bryan, On the beats in the 



[Nov. 24, 



glass all the time, beats will again be heard, showing that the 
nodal meridians do not rotate with the same angular velocity as the 
glass and observer. If the glass be attached to a revolving turn- 
table it is easy to count the number of beats during a certain 
number of revolutions of the table, and it will thus be found that 
the gravest tone gives about 2'4 beats per revolution. As this 
type of vibration has 4 nodes we should hear 4 beats per revolution 
if these nodes were to rotate with the glass, we conclude therefore 
that the nodal angular velocity is in this case about f of that of 
the body. 

It may not, perhaps, be out of place to explain from first prin- 
ciples why the nodal meridians revolve less rapidly than the body. 
Take the case of a ring or cylinder revolving in the direction 
indicated by the arrows in figure 1, and consider the mode of 
vibration with four nodes, B, B, F, H. Suppose also that at the 
instant considered the ring is changing from the elliptic to the 
circular form indicated in the figure. 

Owing to the rotatory motion, the points A, E where the ring 
is initially most bent will be carried forward and parts initially 
less bent will be brought to A and E. Similar remarks apply to 
the points G, G, where the ring is initially least bent. Hence the 
points of maximum and minimum curvature, and therefore, also, 
the nodes must be carried round in the same direction as the ring, 
and cannot remain fixed in space. 





Fig. 1. 



Fig. 2. 



To show that the nodes do not rotate as if fixed in the ring, let 
the small arrows in Fig. 1 represent the directions of relative 
motion of the particles exclusive of the components due to rotation. 
At A, E, the particles are moving towards the centre 0. This will 
of course increase their actual angular velocity and will give them 
a relative angular acceleration in the direction of rotation, as 



1890.] vibrations of a revolving cylinder or bell. 103 

represented by the arrows at A, E in Fig. 2. At C, G the particles 
are moving outwards, and this will retard their angular velocity. 
The particles at B, F are moving with greater total angular 
velocity than the rest; this will increase their "centrifugal force" 
and give them a relative acceleration outwards. Those at D, H 
are moving with the least total angular velocity, and the diminution 
in centrifugal force will give them a relative acceleration inwards. 
Hence the rotatory motion of the mass will give rise to relative 
accelerations of the particles in directions represented by the 
arrows in Fig. 2. If we compare the arrows in Fig. 1 and Fig. 2 
we see at once that the effect of these relative accelerations 
is to cause retrograde motion of the nodes relative to the mass, 
that is, the nodes will rotate less rapidly than the ring. This 
explanation is obviously applicable to all the modes of vibration. 

We will now determine the frequency-equations for the two- 
dimensional vibrations of a thin cylindrical shell or ring of radius 
a which is rotating about its axis with angular velocity co*. We 
shall suppose that the cylinder is also acted on by an attractive 
force /j, times the distance, directed towards the axis. The intro- 
duction of this attraction will enable us to separate the purely 
statical effects of centrifugal force, since by taking /j, = co 2 the latter 
effects will be counteracted. Unless this condition is satisfied, the 
circumference of the cylinder will be in a state of tension. Let 
this tension be T (per unit length of generator) ; and let a be the 
surface density of the cylindrical shell or the line density of the ring. 

When the cylinder is rotating steadily, the condition for relative 
equilibrium gives by resolving normally 

aco a = — \- aua, 

a 

therefore T = aa 2 (co 2 — fi) (1). 

In order to define the position of any point on the cylinder at 
any time t, it will be convenient to employ two systems of polar 
co-ordinates having the centre as pole, in one of which the initial 
line is fixed while in the other it revolves with angular velocity co. 
If, in the undisturbed state the polar co-ordinates in the two 
systems are (a, </>) and (a, 6), we shall have 

cfi = 6 + cot, 

and 6 will be constant for any particle of the ring. 

In the small oscillations, let the small relative tangential and 
radial displacements of the particle be v and w so that its new polar 
co-ordinates are (a + w, </> + vja) or {a + w, d + v/a) in the two 
systems respectively. 

* Compare Lord Eayleigh, Theory of Sound, i. p. 322. 



104 Mr Bryan, On the beats in the [Nov. 24, 

As we shall require to apply the variational equation of energy, 
we must calculate 8?& the virtual work of the effective forces of 
the system. Now if f lf f % denote the transversal and radial 
accelerations of any point, the well-known formulas applicable to 
polar co-ordinates give 

J1 a+w dt ( ' ' 

= v + 2oow, 
f 2 = iv — (a + w)(co+ vfa) 2 
= — aco 2 + w — 2cov — woo 2 
(neglecting squares and products of the small displacements v, w). 
Hence 



8® = fVi^ +/M) aade 
Jo 

= aa \ {— aw 2 8w + (v + 2cow) 8v 
Jo 



+ (w-2cov-wco 2 )8w}8d (2). 

The variation of potential energy will consist of three terms 
representing respectively the work done against the tension T in 
stretching the circumference, the work done against the attracting 
force, and that of bending the cylinder. We shall denote the 
potential energies due to these three causes by W v W 2 and V 
respectively, and their variations will be 8W 1} BW 2 and 8V. 

To find 8W i , let e be the extension produced by the displace- 
ments (v, w) in the arc add. By writing down the stretched length 
of the arc we have 
(1 + ef (add) 2 = dw 2 + (a + w) 2 (d6 + dv/a) 2 ; 

. 2/ dv\ 1/ dv\* 1 (fdw\ 2 a dv 

and therefore, to the second order 



1 / dv\ 
' = a{ W + Td) 



1 {fdw\ 2 _, dv) ,_. 



Hence 



BW^TTSe.ade 

Jo 



by integrating by parts. 



1890.] 
Also 



vibrations of a revolving cylinder or bell. 



105 



W 2 = \ix<r {(a + w) 2 - a 2 } ad6 



i 



sw r 



= I \fxaa (2aw + vf) dO; 
io 

= I [i<ra 2 8wdd + fiaaiv&vdO (5). 

Jo Jo 

Finally for the energy of bending we have if A (1/-R) is the 
change of curvature, 

where, for a cylindrical shell of thickness 2/i 



= (E+T)I. 



SF =?/ 2 '(^ + 1 



and for a ring* 
We readily find 

The variational equation of motion 

becomes, therefore, on slightly rearranging the terms, 
r2w 
0= [aa 2 (- eo 2 + fi) + T] SwdO 

Jo 

+ I \aa(v + 2(ow) Sv + aa (ib — 2coV — w 2 w) Sw 



.(6). 



Tdw . 
a du 



T/dv d 2 w\ 

I ^ — r-^ 1 + fiamu + -g 



£ (d 



+ 1 )w 



8w\ dd. 



a\d6 dd 2 ) ' r ™ ' a 3 \d6 2 

For the undisturbed motion we have v = 0, w = 0, and the first 
line of the above expression gives 

T=<ra 2 (a> 2 -tM), 
as already found (1). 

For the oscillations, we must assume with Lord Eayleigh that 
the extension vanishes to the first order, so that 

dv \ 
d£ 



w = 



and 



o dSv 



.(7). 



* See Lord Eayleigh, Theory of Sound, i. p. 242. Here E denotes Young's 
modulus and //.' Poisson's ratio. 



106 Mr Bryan, On the beats in the [Nov. 24, 

Substituting these expressions for w, Bw and integrating by parts 
the terms containing dSv/dd, we find the equation of motion 

.. d 2 v . dv , „ x d 2 v 

v -de^^de +{( °-^aW 

T ( d 2 v #v\ #_(#_. \* _ n 
+ ^ V d& + m <ra* dF \dd 2 + i ) v - {} - 
To find the frequencies, assume that 

v = A cos (s6 + pt) (8). 

Then the last equation gives (substituting for T its value by (1)) 

(1 + s 2 )p 2 - k<ops = (o> 2 - fi) s 2 (s 2 - 3) + -^ s 2 (s 2 - l) 2 (9), 

therefore 

/ 2sa> \ 2 4a>V , , ,s 2 ( s 2 -S) , /3 s 2 (s 2 - l) 2 

{P-s^Ti) s= (?Ti) 5 + (<B -">-?+!- + ^-F+I--< 10 >- 

> 4a)V , 2 x s 2 (s 2 -3), /3 s 2 (s 2 -l) 2 /n1N 

Let w 2 = -2 — rr-, + (w - h) g , -, + -^4 — n; — =-^...(11), 

(s + 1) v s+1 <ra s +1 v J 

then the two values of p are p v p 2 , where 

2s&) 2,9&) 

and the corresponding motions of the small corrugations relatively 
to the mass are determined by 

V = il COS JS0 + -j— T * + **.*[ ' 

and v=Acos\s0 + s ■ -. t — 'srj 

respectively, together with the relation w = — dv/dd. 

To find the actual motion in space, substitute <j> — oot for 6 in 

the two last equations ; the positions of the corrugations will now 

be referred to the fixed initial line. We find for the two types of 

oscillation, respectively 

{ , s 2 -l j 

v = A cos l scp — £-— y stot + i&jy , 

and v = Acos\s<fi ^— r scot — vrM. 

If the amplitude (A) is the same in both, we see, by addition, that 

their resultant is given by 

/ s 2 — 1 \ 
v = 2A cos'utJcoss (<£ — a-— r tot) (12). 



1890.] vibrations of a revolving cylinder or bell. 



107 



This may be interpreted as representing oscillations of the ring or 
cylinder of period 27r/s3- s having 2s nodes or nodal meridians (where 
dv/dcp = 0) which rotate about the axis with angular velocity 

s 2 -l 

-5 -, CO. 

s 2 + l 

The nodal angular velocity is thus in every case less than co, but 
in the higher tones its difference from co becomes less and less. 

As the nodes are carried round in succession past the direction 
of the observer's ear, beats will be heard, and their number per 
revolution of the material will be 



2.9 



s 2 -l 



sr+r 

The numerical values of these results are tabulated below for a 
few of the smaller values of s : 







Nodal ang. vel. 
Ang. vel. of ring 


Number of beats per 
revolution. 


s 


Number of Nodes. 
2s 


s 2 -l 

s 2 + l 


2*^ 

s 2 + l 


2 


4 


•6 


2-4 


3 


6 


•8 


4-8 


4 


8 


•882 


7005 


5 


10 


•923 


9-231 


6 


12 


•946 


11-351 


7 


14 


•96 


13-44 



The pitch of the intermittent sound is determined by -sr, 
when co = 0, fx = 0, we have 

2 /3 *V-1) 2 



Now 



era 



+ 1 



(13), 



as found by Hoppe* and Lord Rayleigh-f*. Denoting this value 
of -or, by H s , equation (11) shows that if /x = co 2 , so that the 
attraction counterbalances the purely statical effects of centrifugal 
force, we shall have w g 2 > IT/, or the pitch will be somewhat raised 
by the rotation. 

If however fi = 0, and the frequency of rotation is small com- 
pared with the frequency of the vibrations in a non-rotating 
cylinder, so that co is a small quantity of the first order compared 
with II s , equation (11) shows that ot s differs from 1T S by small 
* Crclle, Bd. 63, 1871. t Theory of Sound. 



108 Mr Bryan, On the beats in the [Nov. 24, 

quantities of the second order. This shows that in all cases of 
practical interest the pitch is not perceptibly raised by the rotatory 
motion, and the only noticeable effect is that of the beats already 
described. On the other hand, if the cylinder is revolving very 
rapidly, so that ay is comparable with II s , the vibrations will no 
longer give the effect of beats at all*. 

The results of the last paragraph are very important because 
they admit of extension to rotating shells in general, and afford us 
an easy way of determining the nodal angular velocity and con- 
sequent number of beats per revolution in other slowly-revolving 
systems where the vibrations are not two-dimensional. It will be 
seen that the nodal rotation depends exclusively on B'tB the 
variation of the kinetic energy, and that the square of co may be 
neglected throughout, since it only appears in the expression for 
the frequency, and does not affect the latter to any appreciable 
extent. All that is necessary, therefore, is to calculate #2£. We 
proceed to apply this method to the vibrations which Lord Rayleigh 
has investigated for a perfectly inextensible cylindrical shell of 
length I closed by an inextensible disk at one end*f-. 

Taking the axis of the cylinder as axis of z, and the closed end 
at z = 0, we have, if u denote the longitudinal displacement, 

B*® = dz\ {uBu + (v + 2anu) Bv + (w - 2gm>) Bw} aadd . . .(14), 
Jo Jo 

omitting the terms — ao> 2 Bw — wafBw ; of which the first depends 
on the undisturbed motion, while the second involves small quan- 
tities of the second order. Neither of these omitted terms will 
in any case affect the nodal rotation as they do not contain 
differential coefficients with regard to the time. 
Lord Rayleigh' s conditions of inextensibility are 

du dv du dv _ ,, „ x 

_ = 0, "+33 = 0, 35+« 5 = (15), 

and we suppose Bu, Bv, Bw to satisfy similar conditions. By means 
of the second of these conditions we eliminate w, Bw and have, by 
integrating by parts, 

B® =f l dzf*"iu&u + (v - ^ 2 -4&)S Sv\ aadB (16). 

From the first and third we see that u is a function of 6 and not of 
z, and therefore, that 

_ z du 
a dd' 

* By putting /3=0, yu=0 in (10) we may deduce the solution to the purely 
kinetic problem of determining the oscillations of a rapidly revolving flexible 
endless chain. 

t Proc. London Mathematical Soc. xni., page 5. Proceedings Royal Society, 
Vol. XLV. 



1890.] vibrations of a revolving cylinder or bell. 109 

This enables us to eliminate v, 8v, and therefore by integrating by 
parts we find 

m=s [o dg S7{ U "»("" r^M)) Buaad0 - 

We have not yet assumed the density a to be independent of z. 
Making this assumption and integrating with respect to z we have 

m = fi M -*SS( a -^- 4 » §)} "»•••(«). 

and this shows that the differential equation for u may be written 
in the form 



®-£„i 



Z 2 <f / d 2 il . du\ -r, ( d 2 \ , , 
a 2 W{ U -oW-* CO Td) =F W)( u) > 



the right-hand side containing no differential coefficients with 
regard to the time. Taking u proportional to cos (s<£ + pt) we have 

P 2 + i -2 « 2 (/ + sY - 4ms P ) = F(- s 2 ). 
This may be put in the form 

\p — SH?!*— ; as). 

where -S7 s is a function of s. Comparing this form with (10), the 
corresponding form for the two dimensional oscillations, it is easy 
to see that, in the present case, the nodal rate of revolution will be 

Vs 2 

CO, 



S 2 + W72+1 



3a 2 
ZV 

and the number of beats per revolution will be 

3a 2 



s 2 + S-l 



2s 



2 3a 2 , n 

S+ W 2 + 1 



But we may generalise still further. In any surface of revolu- 
tion, one of Lord Rayleigh's conditions of inextensibility is 

w+ l=° < 19 >- 

If we form #2D, using this and the other two conditions, it is evident 
that we shall arrive at an equation for p of the form 

\p 2 + (1 + s 2 )p 2 — 4<a)sp = a function of s (20), 

VOL. VII. PT. III. 10 



110 Mr Bryan, On the beats in the [Nov. 24, 

where the term \ s p* is derived from the terms uBu which represent 
the virtual work of the effective forces due to the longitudinal 
components (u) of the displacement, and it is important to notice 
that \ can never be negative. Although this last statement is not 
obvious from the variational equation required for the treatment 
of a revolving shell, it becomes evident from the consideration that 
in a non-revolving shell the variational equation leads to the same 
equation for p as the principle of conservation of energy, and that 
in the equation of energy the term containing X s p 2 will arise from 
the kinetic energy of the longitudinal motion (u). This kinetic 
energy is of course essentially positive; or, in other words, the 
whole kinetic energy of the system is greater than it would be if 
the longitudinal motion were neglected. Hence \ is positive. 
The nodal angular velocity, 

and the number of beats per revolution. 

s*+\ -1 



2s 



s* + \ + 1' 



are therefore both greater than they would be if there were no 
longitudinal motion. 

The limiting case is that of a plane circular plate revolving 
about an axis perpendicular to its plane. Here v, w are both zero, 
and the nodal radii are fixed relatively to the revolving mass, the 
vibrations being unaffected by the rotation. The nodal angular 
velocity is therefore a>, and the number of beats per revolution 
is 2s. 

Now in a communication read before the British Association 
at Leeds, I announced that experiments with two different 
champagne glasses attached to a microscopist's turn-table gave 
about 2'6 and 2'2 beats per revolution respectively for the gravest 
tone. While there is nothing contradictory in the former result, 
the latter is too small to be compatible with our theory. As the 
numbers were found by counting the beats during about eight 
revolutions of the table and the mean of 26 observations was 
taken, it is impossible that the discrepancy can arise wholly from 
errors of observation. A further possible source of error was the 
want of uniformity in the angular velocity of the glass. As a 
matter of fact, however, the beats seemed, if anything, most rapid 
when the glass was first set in motion, and as it was not brought to 
rest again during the interval in which the beats were counted, I 
rather doubt this as* the cause of the difference. On the whole I 
should be rather inclined to favour the idea that the discrepancy 



1890.] vibrations of a revolving cylinder or bell. Ill 

is due to Lord Rayleigh's conditions of inextensibility not being 
strictly fulfilled in the neighbourhood of the free edge. 

The results of this paper may therefore be of interest in con- 
nection with the recent controversy on this subject by showing 
how far Lord Rayleigh's theory of thin shells is capable of practical 
application. We see that, if his conditions of inextensibility hold 
good, the number of beats per revolution depends only on the 
shape of the surface and on the law of distribution of density, and 
is in no way dependent on the elasticity of the substance. It is 
therefore readily calculable for a given thin bell. Moreover, the 
number of beats per revolution when the bell is rotated uniformly, 
can easily be counted. If there should be any discrepancy between 
the observed and calculated results which is otherwise unaccounted 
for, this will give us a probable indication to what extent the 
deformation differs from one of pure bending in the bell which is 
the subject of our experiments. 

(2) On Liquid Jets (continued). By H. J. Sharpe, M.A., 

St John's College. 

1. In Vol. VII. Pt. I. I gave an approximate solution of a case 
of liquid flowing from a vessel and becoming a jet, the ultimate 
breadth of the jet being half the diameter of the vessel. I now 
propose to give in detail another case which may perhaps have 
more interest, since in the following the diameter of the orifice 
may be as small as we please compared with the diameter of the 
vessel. In the former solution some ambiguity was attached to 
the position of the orifice. In the present solution there is none. 
Some further observations will be made after the solution has 
been obtained. 

2. We take a case where the outer stream-line (fig. 1) 
AFGBHG cuts the axis of y in a point B such that OB = Ox' the 
semi-breadth of the jet at infinity. 

Let y}r be the stream function on the left of Oy. 
Let OE = 7r, OB = Ccc = ir/p, where p is supposed to be a large 
integer. 

On the left of Oy, let 

— -— = - u = a x e x cos y + a 3 e 3r cos Sy + a 5 e 5x cos hy \ 



+ Xc n ^ nx cos pny + A 
a 3 e 3x sin Sy + a, 
+ Xc n e pnx sin pny 



— -^- = v = a^ sin y + a 3 € 3x sin Sy + a 5 e 5x sin 5y 



(IX 



10—2 



112 



Mr Sharpe, On Liquid Jets. 



[Nov. 24, 



where a lt a s , a 5 , and c n are arbitrary constants, and 2 means sum- 
mation with regard to n from 1 to x . (See Art. 9.) 




oq O 



Then the equation to BHG is 



a % e sin y + la/ x sin Sy + \a/ x sin 5y + 2 -* e*"* sinewy + Ay 

TT 1 . 37T . 57T ^.7T 

= a 1 sin-+X S in-+|a 6 siny + --...(2). 



1890.] 



Mr Sharpe, On Liquid Jets. 



113 



Since Ox' = nrjp we must have 



. 7T , . Sw H - 07T . 

a, sin — I- Icl sin h Aol sin — = 

1 p 6 3 p ° 5 p 



.(3). 



It will be found convenient to replace a t , a 3 , a 5 by other 
quantities such that 

a^^ + A^ a 3 = a 3 + A 3 , a 5 = a b +A t 

When x = 0, we have along OB, on the left of it, 

- u = (a, + Aj) cos y + (a 3 + A 3 ) cos 3y + (a 6 + ^4 5 ) cos 5;^ 

+ Xc n cos pny + A 
v = (a 1 +A 1 ) sin y + (a 3 4- -4 8 ) sin 3y + (a 6 + -4 B ) sin by 
+ Xc n sin pny 

Let % be the stream function on the right of Oy, and on the 
right of Oy let 

— t^ = - u = ^e"* cos y + 6 3 e" 3a; cos 3y + b 5 e~ 5x cos 5y ^ 



.(4). 



(5). 



+ 2,c n 'e~ p " x cos pny + B 



— -^ = v = — b t e x siny — 6 3 e _3a! sin Sy — b h e~ bx sin 5y 



(6). 



dx 



— 2c„'( 



sin pny 



.(7). 



(8). 



It will be found convenient to put 

When x = 0, we have along OB, on the right of it, 

- m = (a, - A t ) cos y + (a 8 - .4 3 ) cos 3# + (a s - 4 6 ) cos 5y 
+ 2c/ cos pny + B 

»=-(«,- A) sin y-i%- A 3 ) sin %/ - ( a s - A) sin 5 2/ 

-2c B 'sinpM/ 

By Fourier's Theorem, suppose we have from y = to 7r/p 
2A t cosy+ 2J. 3 cos Sy + 2.4 5 cos 5y = Q + 2q n cos pny... (9), 
which will be true at both limits. 

Identifying the first of (5) and the first of (8) we have 

C n ~C n ' + q n = (10), 

A-B + Q = (11). 

Again, suppose we have from y = to ir/p 

2^ sin y + 2a 3 sin Sy + 2a 5 sin 5y = %r n sin pny . . . (12), 



114 Mr Sharpe, On Liquid Jets. [Nov. 24, 

which will be true at both limits if 

. 7r . Sir . 5tt _ ... ox 

a, sin- + a, sm f-a K sin — = (1*3). 

1 p 3 p 5 p 

Identifying the second of (5) and the second of (8) we have 

c n + c n ' + r n = (14). 

From (6) the equation to AFGB is 

c ' 
\<f x sin y + ±b 3 €~ 3x sin Sy + £<f" sin 5y + % -^ e mx sin W + % 

7 . 7r , 7 37r ., . 5tt Bit /1c . 

= 6.sui- +16, sin— +46. sin — + — (15). 

1 p 3 3 ^,55 ^ p 

Since J.« = 7r, we must have 

, . 7T 17 . 37T 17 . 57T (»— 1)2?7T / - „>. 

6, sm - + 46, sm — + 46. sin — = ^ '- ... (16). 

p p p p 

It has already been shewn that there is a sharp turn at F. 
From (9) we have 

Q = 2 ^sin? + 2^f sin^+2^#-sin 5 ^...(17). 

^ 1 1T p 3 37T p 5 57T p 

, p . 7T COSW7T . . 3» . 37T COS 717T 

#„ = — 4 J.J - sm — x -2-g — t - 4J. S — sm - - x -^-5 — ? 



7T 



p p 2 n 2 — 1 3 7r ^> pn* — 9 



. . 5« . 57T COS 717T /1C ,. 

- 4<A, -^- x sm — x -^-2 — — (18). 

5 7T p pV-25 v y 



From (12) we have 



P . 7T 71 COS 727T , JO . 3*7T W COS W7T 

r = — 4a ■£- sm - x — »-= — =- — 4a„ — sm — x — 2-5 — ~ 
71 * 7r p pn - 1 7r £> pn — 9 

. » 2 . 57T ?l COS W7T ,, AX 

-4a. *- sm — x , g a , (19). 

From (10) and (14) c n = - %r n - %q n \ , 9m 

c ' = _i r + i a f ^ u '" 

°« 2'n^ 2^/J 

When the above conditions are satisfied, the velocities on each side 
of OB will be continuous. 

3. Equations (3), (13) are the only equations connecting the 6 
quantities a 1} A v a s , A 8 , a B , A B , therefore so far 4 are independent. 
Equations (11), (16), (17) serve to connect A with B and either 
with ctj, A v &c. From (20) c n , c n ' are functions of n and a v A x , 
&c. The 6 quantities * x , A x , &c. may any of them (consistently 
with the above relations) be as small as we like, but none of them 



1890.] Mr Sharpe, On Liquid Jets. 115 

will be taken so large as to be comparable with p. We will sup- 
pose p so large that 

p . it p . Stt p . 5tt 
- sin - , £— sin — , -f- sin — 

7T p 6lT p 07T p 

are very nearly unity. 

For some investigations connected with this subject it might 
be necessary to expand these expressions in powers of 1/p, but 
for our present purpose it will suffice if we suppose them actually 
unity. To this end we must suppose p to be at least 22 (this 
number making the largest rejected term ^th of what is retained), 
but we may have to take p much larger than this. 

(3) and (13) then become 

a t + a s + a 5 = ,(21), 

a 1 + 3a 8 +5a 6 = (22). 

(16) and (17) become 

K + h + b s =pB (23), 

Q = 2A y + 2A 3 + 2A 5 (24). 

From (11), (21), (23), (24) we readily get 

A =pB = -Q=- 2A t - 2A 3 - 2A S (25). 

We will next suppose p so large that we may safely expand 
the fractions in (18) and (19) in ascending powers of 1/p. To 
this end p must be at least 50, and we shall get, retaining only 
the most important terms in q H and r n , 

4C0S«7T , . _ . ar , . N ,„_. 

g, = - v (A 1 + 9A a +25A B ) (26). 

4cos?i7r, aH -,-,- x /«h,\ 

For large values of p, r n vanishes compared with q n . We 
shall therefore retain only q n in equations (20). We shall further 
suppose a v a 3 , a 5 to be small quantities multiples of 1/p 2 and that 
their ratios are such that 

a 1 +3a 8 + 5a 6 = ..(28). 

(The object of these assumptions will be presently seen.) 

From (21) and (28) therefore we may put 

-a i = ia a = -a 5 =p 2 (29), 

where p 2 is a small quantity supposed to be a multiple of 1/p 2 . 



116 Mr Sharpe, On Liquid Jets. [Nor. 24, 

It will then be found from equations (4), (22) and (29), that 
so far we may regard a t and a 3 as arbitrary, and A x , A s> A b , a 5 as 
determined in terms of them by means of the following equations: 

A = - a i-P Z ' A 3 = - a 3 + 2 P2> A 5 = i( a i + Sa s)-P2\ /CAN 

« 5 = -*K + 3« 3 ) }"^> 

We shall further get, omitting a small term multiplied by p 2 in 

the bracket following, 

, 4 cos nir /a . « \ /oi\ 

o n = -c n '= v (2« 1 + 3« 3 ) (31), 

A=pB=%cc 1 + j.ct 3 (32). 

4. We shall now make BHC as far as possible a line of con- 
stant velocity. In (2), putting for shortness z for e x and modifying 
by means of (29) and (31), we get for the equation to BHC 

-i(2 ai + 3« 3 )S^^sin W ...(33). 

It is obvious that for any point on BHC the coefficient of p 2 
cannot exceed f and that the S cannot exceed % (l/^ 3 ) which is 
about 118. All along BHC therefore y differs from ir/p only by 
quantities at most of the 2nd order of smallness. (They are in 
fact of the 3rd order.) In (1) therefore we may put unity for 
cos y, cos Sy and cos 5y, and then modifying (1) by means of (29) 
and (31), we have at every point on BHC 

-u=p 2 (-z+2z s -z 5 ) 

+ i (2a, + 3* 3 ) 2 ^-^ *- cos W + A . . .(34). 

Jj lb 

From (1) and (31) and reasoning like the preceding, it is evi- 
dent that v is at most of the 2nd order of smallness, because c n is. 
Therefore (w 2 + v 2 ) only differs from A 2 by quantities of the order 
1/p 2 . Practically therefore at every point on BHC (34) gives the 
whole velocity. This velocity consists of a constant part A and 
a variable part. A portion of this variable part can be made, if 
we like, to vanish by putting 

2a 1 +3a 8 = (35). 

The remaining portion will be a maximum or minimum at a 
point determined by 

-z+6z 3 -5z 5 = (36). 

This is satisfied by z = 1 or the point B. 



1890.] Mr Sharpe, On Liquid Jets. 117 

If p 2 is positive, we readily see that a maximum has been 
obtained. 

From (1) and (28) v vanishes at B. From (33) we see that 
BHG is always above its asymptote. When z 2 = i (36) furnishes a 
minimum, which, judging from the equation of continuity, should 
give us a maximum ordinate, which we indicate by H in the 
figures. This idea is corroborated, as we can prove from (1) that 
z 2 = i- causes v to vanish. 

If p 2 is negative, the words " maximum and minimum " must 
be interchanged in the above sentences. BHG is then always 
below its asymptote, and at H there is a minimum ordinate or 
Yena Contracta (fig. 2). 

In both cases, as we are not so much concerned with the sign, 
as with the actual magnitude of the error in the velocity derived 
from (34) we see, since the coefficient of p 2 in (34) vanishes when 
z — 1, that the error is of the 3rd order of smallness at B and of 
the 2nd order at H, as far as p is concerned. 

We now proceed to trace the curve AFGB on the right of Oy. 



From (7), (30) and (35) we get 



.(37), 



also from (32), pB = i|a 1 

therefore from (15) the equation to AFGB becomes, putting z for 

16a 

^-i ( y - ,r) = - (2a x 4- p 2 ) z sm y 

+ (fa + ti> 8 ) z* s^ % - (Jte + ip 2 ) z* sin 5y. . .(38). 

As p 2 is small compared with a x we see that it makes little differ- 
ence whether we put +p 2 or — p 2 in the last equation, as the 
curves obtained will only differ slightly in shape and position. 
In fact, to get a general idea of the form of the curve, we may of 
course practically put p 2 = 0. 

We thus get for the abscissa of G where the curve cuts the 
line y = irjp 

T5 ^ + r ?' 
whence z = "628 if x is about £. 

At F u = 0, so we get from (6) for the abscissa of F 
0— jr + f^-^jP+A, 

EF therefore increases slowly as p increases (see Art. 8). If for 
instance p = 100, EF = 5^ nearly. 



118 Mr Sharpe, On Liquid Jets. [Nov. 24, 

To consider more carefully the form of the small portion GB. 
As in this portion y < wjp we may put in (38) y, 3y, hy for sin y, 
sin 3y, sin oy. We can then readily shew that in passing from the 
curve given by + p 2 to the curve given by — p 2 we alter the 
ordinate of GB by a quantity at most of the 3rd order of smallness*. 
It is interesting to compare with this the fact that the corresponding 
alteration in the ordinate of the jet at all points (except in the 
neighbourhood of B) is of the 2nd order of smallness. Thus a 
very small alteration in the vessel produces a disproportionately 
large alteration in the jet, in fact changing a maximum into a 
minimum ordinate or vice versa. 

The Problem thus seems to suggest a point of contact with 
Lord Rayleigh's article " On the Instability of Jets " — given in 
Vol. x. of the Proceedings of the London Mathematical Society, 
though it must be admitted that in that article the author is con- 
sidering not the effect of the vessel on the form of the jet, but the 
instability of the jet itself due to capillary force. 

6. Of course there is nothing whatever to prevent us putting 
p 2 = in Art. 4, in which case we see from (34) that not only at B, 
but all along the jet the error in the velocity would be at most of 
the 3rd order of smallness. 

We may remark that no attempt has been made to draw the 
figures to scale, as that would be difficult. 

7. Supposing p 2 not to be zero, and the linear unit of measure- 
ment to be comparable with 1 foot (or perhaps better say 2 / or S f ), 
it would approximately make no difference to the solution if a 

* From (38) we have in curved portion GB (putting a for a x for shortness), 

j^ (V - t) = - (2a +p 2 ) zy + (g* + 2 Pz J ^hj - (^ a +p 2 J z^y. 
Now keeping z constant, suppose we put -p 2 for +p 2 and let y become y x then 
jg^ (Vi ~ t) = ~ (2« -p 2 ) zy x + ( g a ; - 2pA zhj x -^a-pA z 5 y x ; 
.'. subtracting, we have nearly 

is? (y - 2/1) = (y - 2/1) [ - 2az + 3 azS - 5 a * 5 J + 2 y [ -v& + 2 ^ z3 -iv 5 ]- 

As in GB z<-628, 

we have approximately 

{y-y 1 )[-2ax'628 + &c.]=-2yp a x -628; 

.-. as in GB y < — . 

y -y 1 is at most of the order -= . 



1890.] Mr Sharpe, On Liquid Jets. 119 

constant accelerating force g were acting on the fluid parallel to 
yO, for we see from (33) and (34) that in that case the equation 
u 2 + 2gy = constant would be satisfied accurately to the 3rd order 
of small quantities all along BUG, as the coefficient of p 2 in (33) 
is of the order 1/p. We thus get a sort of millrace BUG, and if p 
be large enough, AF might almost be regarded as a free surface. 

8. It was proved in Art. 5, in the solutions already obtained, 
that EF is a function of p which increases slowly as p increases. 
Suppose it were required to find a solution wherein EF is a con- 
stant quantity independent of p, but not zero. We could do so by 
introducing into equations (1), (6) &c. additional terms involving 
(say) sin 7y and cos 7y with 2 additional arbitrary constants. The 
whole of the above process would then have to be gone through 
and it would be found quite possible to satisfy the new conditions. 
p would then have to be >70. The order of the errors would 
of course remain unchanged. In this case there is no point G, but 
the outer stream-line, after touching the asymptote at B, goes up 
to F. The figure on the right of Oy would then have a much 
closer resemblance to a cistern with an orifice at the bottom or (as 
we may suppose the figure symmetrical with regard to the axis of 
x) in the middle. There would be found to be an infinite number 
of such solutions. It must however be observed that we cannot 
find a solution of this kind wherein EF is actually zero. To prove 
this we will first consider the case treated in Arts. 1 — 6 above. 
Since from (31) and (35) c n ' — 0, we have from (6) to determine the 
abscissa of F, 

0=-b 1 z-b/-b 5 z 5 + B. 

If this is satisfied by z = 1 Ave must have 

= -b 1 -b a -b 5 + B (39). 

But from (16), for large values of p, 

b t + b t +b B = Bir (40). 

(39) and (40) being incompatible, EF cannot be zero. An exactly 
similar proof would apply if we introduced into (1), (6) &c. ad- 
ditional terms involving (say) sin 7y, cos ly. 

9. In equations (1) (6) &c. the first three terms on the right- 
hand side involve the three odd numbers 1, 3, 5. We are not 
obliged to use odd numbers. Any integers will give a distinct case, 
provided that p is large compared with the largest of them. If we 
want a case which is compatible with the lowest possible value of 
p, we should choose the numbers 1, 2, 3, and then p would be as 
small as 30. So in Art. 8 if we chose the numbers 1, 2, 3, 4 we 
could have p as small as 40 and yet have EF to a considerable ex- 
tent independent of p. 



i20 Mr Brill, On the Application of Quaternions [Nov. 24, 

(3) Note on the Application of Quaternions to the Discussion 
of Laplace's Equation. By J. Brill, M.A., St John's College. 

1. The object of the following communication is to obtain, as 
far as is possible, with the aid of the Calculus of Quaternions, a 
theory for the three-dimensional form of Laplace's Equation 
analogous to the well-known theory of Conjugate Functions, which 
has proved of so much service in the treatment of the two-dimen- 
sional form. 

The two related solutions of the two-dimensional theory are 
replaced in the three-dimensional theory, not by three, but by 
four related solutions. Thus if a, ft, 7, & be four quantities con- 
nected by the equations 

d8_d_y_dft 

dx dy dz ' 

98 da dy 

dy dz dx ' 

d8_dft_da 

dz dx dy ' 

dx dy dz ' 

then it is easily verified that a, ft, 7, 8 are severally solutions of 
Laplace's Equation. Moreover, if we interpret 8 as the velocity 
potential of a case of irrotational fluid motion, then a, ft, 7 are 
the components of the vector potential which occupies a similar 
place in the three-dimensional theory to that occupied by the 
current function in the two-dimensional theory. 

If we now write 

r = — 8 + ia +jft + k<y, 
we have 

_ _/ck dft dy\ .(_<$ d J_d0 
\dx dy dz) \ dx dy dz 

.( dB , da dy\ '( d8 dft da\ n 
+ A-dy + d-z-i) +k [-dz + £-dy)-° (1 > 

Further, let 

p = — 2ix + jy + kz, 

a = ix — 2jy + kz, 
t = ix + jy — 2kz. 
Then we have 

Vp = Vo- = Vt = 0, 



1890.] to the Discussion of Laplace's Equation. 121 

and therefore p, a, t are vector solutions of equation (1); but, 
since p + <r + t = 0, 

they only supply us with two independent solutions. This, how- 
ever, will be sufficient for our purposes, since for the development 
of the theory we only require to know two independent special 
solutions. There is a certain disadvantage about these forms as 
the selection of two of them renders the work unsymmetrical. 
There is a symmetrical quaternion solution of (1), which involves 
x, y, z linearly, viz. 

2 (x + y + z) +i(y - z) +j (z - x) +k (x -y); 

but 1 have not been able to hit upon a second symmetrical 
solution involving x, y, z linearly. 

2. We are now in a position to shew that there exists a 
relation of the form 

dr = dp.R + da.S (2), 

where R and S are quaternions whose expressions involve x, y, z, 
but not dx, dy, dz. To verify this, assume 

R = u + if+jg + kh, 

S = v + il +jm + hi ; 

and then equation (2) becomes 

— dS + ida +jd/3 + kdy = (— 2idx +jdy + kdz)(u + if+jg + kh) 

+ (idx — 2jdy + kdz) (v + il +jm + kn). 

This involves the existence of the four relations 

-dB = (2/- I) dx + (2m -g)dy- (h + n) dz, 

da = (v — 2u) dx + (h — 2n) dy — (g + m) dz, 

d/3 = (2h -n)dx + (u -2v)dy + (/+ I) dz, 

dy = (m - 2g) dx + (21 -/) dy + (u + v) dz; 

and since these are to be satisfied independently of the values of 
the ratios dx : dy : dz, we obtain the following twelve equations : 



™-l-2f 


dy= g - 2m ' 


dB , 
dz =h + n > 


da 

— = v — 2u, 
ox 


— = h — 2n, 
dy 


da 


dx- = 2h ~ n > 


7T- = u — 2v, 
dy 


£*/+< 


d £ ==m - 2 v> 


dy U * 


dy 



122 Mr Brill, On the Application of Quaternions [Nov. 24, 

As we have here twelve equations, and only eight quantities 
to be determined, it is obvious that the equations imply four 
relations between the differential coefficients of a, /3, 7, B. It is 
to be remarked that the group of twelve equations consists of 
four sets containing three equations each, the three equations of 
any one set containing only a single pair of the above-mentioned 
eight quantities. The pairs are as follows : %i and v, f and I, g 
and m, h and n. Thus each set of three equations will furnish 
us with a single relation, and enable us to determine, subject to 
that relation, the values of a pair of the eight quantities required. 

The four relations connecting the differential coefficients of a, 
/3, 7, S are easily shewn to be identical with those contained in 
Article 1, and consequently the existence of equation (2) is justi- 
fied. Thus we have : 

da dy , d8 

= ^ L = — q — m — m + 2g = g-2m = ^-, 

dz dx y y J dy' 

dB da n7 7,0 7 , 98 

dx dy oz 

ox ay oz 

It is also easily proved that the values of R and 8 can be 
expressed by the formulae 

« B -(*4-*l>- M -(4 : -*s)-- 

3. Let p and q be two independent quaternion solutions of 
equation (1), then according to the preceding Article we have two 
equations of the form 

dp = dp.T + d<r . U, 

dq = dp . V+ da . W. 

From these we obtain 

dp . U' 1 = dp . TU~ l + da, 

dq. W~ l = dp.VW- x +d<r; 

whence by subtraction 

dp . U' 1 - dq . W- 1 = dp (TU' 1 - VW' 1 ), 

and therefore 

dp = (dp . U~ x - dq . W- 1 ) (TU- 1 - VW'T- 



1890.] to the Discussion of Laplace 's Equation. 123 

Similarly we should obtain 

da = (dp . T- 1 - dq . v~ i ) ( ut~ i - wv~ l y\ 

Substituting these two values for dp and da in equation (2), we 
see that it takes the form 

dr=dp.P + dq.Q (3), 

where 

P=U~\ (TU~ l - VW 1 )- 1 .R + T-'.iUT' 1 - WV' 1 )-' . 8, 
and 

Q = W' 1 . (VW- 1 - TU'Y .R+V' 1 . (WV 1 - UT' 1 )- 1 . s. 

4. It now only remains to remark that equation (3) is to 
be regarded as the three-dimensional analogue of the relation 

dw =f (z) dz, 

where w=f(z)=f(x + iy); which relation expresses that the 
ratio dw : dz depends only on the origin from which the vector 
dz is drawn, and not upon its direction. If we have two complex 
variables u=oc + cy and v = z + it, and if w =f(u, v), then we have 
a relation of the form 

dw = ~- du + ~- dv, 
ou ov 

where the values of df/du and df/dv depend only on x, y, z, t 
and not upon the values of the ratios dx : dy and dz : dt. 

In the quaternion theory the order of factors in a product is 
material, and it turns out that the differential factors must be 
placed before the finite ones. 

The existence of the relation expressed by equation (3), seems 
to point to the necessity of a discussion of quaternion functions 
of two variables, i.e. involving two variable quaternions. This 
I hope to be able to furnish in a future communication to the 

Society. 

I am not at present prepared to give a geometrical inter- 
pretation of equation (3). Our work furnishes us with materials 
for calculating P and Q in terms of the differential coefficients 
of the elements of p, q and r, but the working out of the values 
by this method would be very tedious. It is possible that when 
a geometrical interpretation is discovered for equation (3), this 
may suggest some shorter method of obtaining the required 
values. 



124 Mr Brill, On the Application of Quaternions [Nov. 24, 

5. In conclusion, we may notice that Laplace's Equation is 
a particular case of a more general equation to which the qua- 
ternion method is applicable, viz. the equation 

d\i d 2 u d\i ^ u _o 
dtf + dy i+ d7 + d?~ 

In this case, as in the former, we have four related solutions, 
but the equations connecting them are 

8S_<57_8/3 da 
dx dy dz dt ' 

dy dz dx dt ' 
dS_d/3_da dy 
dz dec dy dt ' 

dx dy dz dt 

These four equations are equivalent to the single quaternion 
equation 

{st + i l + ik +k s) < ~ S + h+j0 + h) = ° (4> 

We however require three special solutions of this equation, for 
which we may take the following : 

u =t — ix+ jy + hz, 

v = t + ix — jy + hz, 

w = t + ix + jy — hz. 

And, proceeding as in Article 2, we obtain a relation of the 
form 

ds = du . U+ dv . V + dw . W, 

where s is any solution of (4), and IT, V, W are quaternions whose 
values depend upon x, y, z, t and not upon dx, dy, dz, dt* From 
this we can deduce by proceeding as in the former case that if 



We have 



^-(»+*5>--('B + >S)- 



1890.] to the Discussion of Laplace s Equation. 125 

f„ q and r be any three independent solutions of (4), then we 
obtain a relation of the form 

ds = dp . P + dq . Q + dr . R. 

[Since the paper was read I have obtained an analogue for 
another fundamental theorem concerning functions of a complex 
variable, viz. the theorem that 

\f(z)dz=Q, 



I 



the integral being taken round a closed curve not involving any 
singular points. 

If we integrate over a closed surface, we have 

I Uidydz +jdzdx + kdxdy) r = j J IdxdydzVr = 0. 

This theorem will of course require corrections similar to those 
necessary to make the other theorem general, but the material 
for furnishing these is ready to hand. It is to be noticed that 
in the statement of this theorem, as in that of the theorem of 
Article 3, the infinitesimal factor has to be placed before the 
finite one. 

There is also a corresponding theorem for the case discussed 
in Article 5, which may be written in the form 

I \\(dxdydz + idydzdt +jdzdxdt + kdxdy dt) s = 0.] 

(4) On a simple model to illustrate certain facts in Astronomy, 
with a view to Navigation. By A. Sheridan Lea, Sc.D., Gonville 
and Caius College. 

The model consists of a small solid sphere, representing the 
earth, placed in the centre of a hollow sphere composed of circles 
of wire. Of these, one represents the celestial equator, one the 
ecliptic, and the others various meridians corresponding to the 
parallels of longitude on the earth. Small coloured balls can be 
attached at any point on the wire circles to represent at any time 
the positioDs of the sun or any star relatively to the earth. A wire 
representing the axis of the ecliptic can be attached to one of the 
vertical meridians, and this carries the moon with the axis of her 
orbit inclined at 5° to that of the ecliptic and movable round the 
latter. The model of the earth is perforated by holes bored at 
right angles to its surface by means of which a movable horizon, 
carrying a wire at right angles to it which determines the zenith, 
can be attached at any point of the earth. The model whi'e 

VOL. VII. pt. in. 11 



126 Mr Burnside, On a paper relating [Nov. 24, 

demonstrating the relative movements of the earth, sun, moon ana 
stars is more particularly intended to illustrate and explain, ohe 
astronomical observations by means of which the position of an 
observer is determined on the earth's surface, e.g. meridian and 
ex-meridian altitudes of the sun or a star whether the latter be 
circumpolar or not. It of course affords a clear explanation of the 
more important terms used in navigational astronomy, serves also 
to illustrate the cause of the varying length of day and night at 
different seasons of the year, the phases of the moon and their 
relation to the tides, and affords a rough demonstration of the 
course of solar eclipses. 

(5) Note on a paper relating to the Theory of Functions. By 
W. Burnside, M. A., Pembroke College. 

In a paper " on the geometrical interpretation of the singular 
points of equipotential curves " printed in Vol. VI. of the Society's 
Proceedings, Mr Brill has stated some properties of algebraical 
equipotential curves which are of very doubtful accuracy. 

The point of view taken seems to be this : that the several 
members of an equipotential family of curves do not generally 
meet in real points at all, and that when they do the points of 
intersection of the consecutive curves of the family are the branch 
points of the function which gives rise to the family. 

It is not explicitly stated, though it seems to be implied, that 
these points, viz. the branch points, are the only real points of 
intersection of such a family. Several of the results obtained in 
the paper in question depend on the accuracy of the above state- 
ments, so that it is perhaps worth while to examine them in 
some detail. 

For this purpose I limit myself to the case in which not only 
the curves themselves are algebraical, but also the function which 
gives rise to them. I also consider, except in the last paragraph, 
real points on the curves and real points only. 

Suppose that f{z, w) = is an equation of the nth degree in 
w. If oc + iy and u + iv are written for z and w, two equations 
each connecting x, y, u, v will result from f— 0, and by eliminating 
u and v alternately between these the equations of two systems of 
algebraical curves will be obtained of the forms 

fx 0. y> u ) = °> / 2 0> y> v ) = °- 

The clearest conception of the intersections (real) of these curves 
may be obtained in the following manner. 

The function w of z, which as defined by the above equation is 
w-valued, can be represented as a one-valued function on the n- 
sheeted Riemann's surface belonging to the equation /=0; and 



1890.] to the Theory of Functions. 127 

therefore at every point of this surface u and v will each have a 
single definite value, the only exceptions being the points 
corresponding to infinite values of w, at which (since w is an 
algebraical function) u and v will each take all possible values. 
If then on this surface the curves u = c(— oc < c < oo) are drawn 
the separate curves will nowhere meet each other except at the 
points where w is infinite, and through these points all the curves 
of the family will pass. The same remark applies to the w-curves. 
The Niemann's surface on which the curves have been drawn 
consists in its simplest form of n superposed infinite planes, the 
continuity between different planes being provided for by cross- 
cuts (to use Clifford's translation of " Verzweigungsschnitte ") con- 
necting properly the branch -points. If now these n planes be 
regarded as transparent so that the curves may be seen as though 
lying in one plane, the result will be the same as though they 
had been so drawn originally. It is then at once clear that 
through every point of the single x, y plane n it-curves and n 
v-curves will pass. [It is only by properly taking the curves in 
pairs so that u + iv is a root of f{z, w) = for the value of x + iy 
considered that the u- and ^-curves will cut at right angles.] 
The only exception will be that through the points x + iy, which 
make one or more values of w infinite, all the curves of both 
systems will pass. The branch-points, i.e. the points x + iy which 
make the equation f= have equal roots, will be distinguished 
in this way; that whereas generally the values u v u 2 ...u n of the 
parameters of the n w-curves passing through a particular point 
are all different ; if the point is a branch-point at which r roots 
of the equation f= become equal, r branches of one curve u t 
and n — r other curves u r+1 ,...u n will pass through the point; as 
well, of course, as r branches of a curve v 1 and r other curves 
v _,_,... v . It is obvious at once that such a family of curves can- 
not have what is usually called an envelope (real), for this would 
clearly necessitate the Riemann's surface consisting of an infinite 
number of sheets. 

The point in which Mr Brill seems to have gone astray is the 
following. In § 3 of his paper he says that if the curves v and 
v + dv intersect, the necessary condition is 

/ { u + iv] =f [u + i (v + dv)} ( A) ; 

[the equation between z and w in the paper referred to is written 

This clearly is not necessary, for the curves will intersect if 

f{u + iv}=f{u' + i(v + dv)} (B), 

where u and ii are different. 

11—2 



128 Mr Burnside, On the Theory of Functions. [Nov. 24, 1890. 

In the light of the above geometrical reasoning the condition 
(A) is, except at a branch-point, equivalent to supposing that the 
imaginary part of w has the two different values v and v + dv at 
one point of the Riemann's surface, which is inconsistent with w 
being a single-valued function on the surface (unless the point is 
an infinity of w). 

From the condition (A) Mr Brill at once deduces the relation 
f'(ti + iv) = 0, 

on which the statements mentioned at the beginning of this note 
are based. 

The condition (B) implies that the imaginary part of w has the 
two values v and v + dv at points corresponding to the same x + iy 
on two different sheets, say 1 and 2, of the Riemann's surface. The 
values of v on each sheet are continuous and hence on sheet 2 
there must be a branch of the curve v indefinitely near the branch 
v + dv. This branch v on sheet 2 and the branch v above con- 
sidered on sheet 1 will intersect, generally at a finite angle, when 
the curves are drawn on a single plane ; and hence the locus of 
real intersections of consecutive curves, which it was shewn above 
could not be an ordinary envelope, is a locus of double points. 
It is to be noticed that corresponding to the double point con- 
sidered on the v-curve there will generally not be one on a it-curve. 

Mr Brill applies his equation 

f (u + iv) = 

to discuss not only the real but the imaginary intersections of the 
curves in question. To this purpose the equation appears to me 
to be entirely inapplicable. 

In considering the imaginary intersections of two ^-curves, 
the problem in hand becomes one of functions of two complex 
variables defined by equations like 

f t 0, V, u) = 0, 
where x and y may both be complex. 

If the attempt be made to deal directly with imaginary values 
of x and y, say x t + ix 2 , y t + iy 2 in the original equation defining 
the function, £ becomes x x — y 2 + i (^ 2 + 2/i), and in the analytical 
work all trace of the original point is lost, as it is impossible to 
pass back from the two real quantities x x — y 2 and x 2 + y x to the 
original complex quantities which gave rise to them. 



COUNCIL FOR 1890—91. 

President. 
G. H. Darwin, M.A., F.R.S., Plumian Professor. 

Vice-Presidents. 

John Willis Clark, M.A., F.S.A., Trinity College. 
C. C. Babington, M.A., F.R.S., Professor of Botany. 
G. D. Liveing, M.A., F.R.S., Professor of Chemistry. 

Treasurer. 
R. T. Glazebrook, M.A., F.R.S:, Trinity College. 

Secretaries. 

J. Larmor, M.A., St John's College. 
S. F. Harmer, M.A., King's College. 
E. W. Hobson, M.A., Christ's College. 

Ordinary Members of the Council. 

J. W. L. Glaisher, Sc.D., F.R.S., Trinity College. 

W. N. Shaw, M.A., Emmanuel College. 

W. Gardiner, M.A., Clare College. 

W. Bateson, M.A., St John's College. 

A. Cayley, Sc.D., F.R.S., Sadlerian Professor. 

W. J. Lewis, M.A., Professor of Mineralogy. 

W. H. Gaskell, M.D, F.R.S., Trinity Hall. 

E. J. Routh, Sc.D., F.R.S., Peterhouse. 

A. Hill, M.D., Master of Downing College. 

A. S. Lea, Sc.D., F.R.S., Gonville and Caius College. 

A. Harker, M.A., St John's College. 

L. R. Wilberforce, M.A., Trinity College. 



PROCEEDINGS 



OF THE 



Citmbritrgc pjilasopljtnil jftofrietj. 



January 26, 1891. 
Prof. G. H. Darwin, President, in the Chair. 

The following Communications were made to the Society : 

(1) On the Electric Discharge through rarefied gases without 
electrodes. By Prof. J. J. Thomson. 

A vacuum tube was exhibited in which an electric discharge 
was induced by passing the discharge of Leyden jars through a 
thread of mercury contained in a glass tube coiled four times 
along it. The induced discharge was found to be confined to the 
part of the vacuum tube which was close to the primary discharge, 
and it did not shew striae. 

It was also demonstrated that an ordinary striated discharge 
is strikingly impeded by the presence of a strong field of magnetic 
force. 

(2) The Laws of the Diffraction at Caustic Surfaces. By J. 
Larmor, M.A., St John's College. 

1. One of the most striking phenomena in connection with 
the propagation of light or other undulations is the circumstance 
that under certain conditions, common in optics and easily realis- 
able in the case of superficial water waves with varying depth of 
water, there exists a geometrical boundary beyond which the 
undulations cannot penetrate at all, but in the neighbourhood of 
which the disturbance is very much intensified. 

In the first approximations of Geometrical Optics, where the 
undulations are treated as a system of rays, and the energy is 
considered to be propagated along them, the caustic or envelope 
of the rays appears as a surface of infinitely great concentration 
of energy, which is also the boundary of the space into which the 
energy can penetrate. The conception that must replace this in 

VOL. VII. PT. iv. 12 



132 Mr Larmor, On the Laws of the [Jan. 26, 

a more exact view of the phenomena is that of the theory in- 
vestigated by Sir George Airy for the case of the rainbow, on 
the basis of Fresnel's theory of diffraction. It was shown by 
him that, outside the real caustic of maximum concentration, 
the energy of the undulations gradually fades away, so that with 
very minute wave-lengths the boundary of the caustic is quite 
sharp ; but that inside the caustic there is presented a series of 
successive maxima and minima, in bands running parallel to the 
absolute maximum or caustic surface. 

As the calculations of Sir George Airy had reference chiefly 
to the phenomena of supernumerary rainbows, he only cared to 
obtain the relative distances and illuminations of the succession 
of bands along the asymptote of the caustic. 

But the peculiarity of this case of diffraction is that there is 
no question of an aperture limiting the beam of light, so that the 
degree of closeness and other relations of the bands must depend 
only on the character of the caustic surface itself, along which 
they run. The law, connecting these elements, which is thus sug- 
gested for investigation, comes out to be very simple. It appears 
that for homogeneous light the system of bands is similar to 
itself all along the caustic, as regards relative positions and 
relative brightness, and that they are therefore similar to the 
supernumerary rainbows calculated by Airy and verified experi- 
mentally by W. H. Miller; while the absolute breadths at different 
parts vary inversely as the cube root of the curvature of the 
caustic surface along the direction of the rays. For different 
kinds of light the breadths vary as the wave-length raised to the 
power two-thirds. These laws are exact for the first few bands, 
usually all that are visible, owing to the extreme closeness of the 
subsequent ones; they form in fact the physical specification of 
the nature of caustic surfaces. 

2. These statements will be verified in the course of the 
following analysis of the diffraction near the surface of centres of 
a wave front, which forms the natural extension of Sir George 
Airy's investigation for that portion of the caustic which sensibly 
coincides with its asymptote. 

In the first place, taking a cylindrical wave-front, and referring 
it to the tangent and normal as axes, we have for its equation in 
the neighbourhood of the origin 

z = ax 2 + bx 3 + 

The inclination of the tangent at the point x is <£ = 2ax, the 
radius of curvature is 

R = O'llV = 1 ( x _ ^ x 
\dx 2 / 'la V 2a 



1801.] Diffraction at Caustic Surfaces. 133 

and the radius of curvature of the evolute or caustic is 

_dR = Sb 
P ~dcf>~ (2af 

We have to determine the disturbance, at a point (£, l/2a) in 
the focal plane of the origin, due to the propagation of this wave. 
If the amplitude of the motion in the wave-front is t sin 2tt£/t 
per unit length, the value required for the point in question will 
be 

where for the part in the neighbourhood of the origin 

«-{«-7+(iH? 

= 7~* [1 " 7"f« + *7 4 ?V - i (j'ba" + 7 4 f + yT) «'+••■) 
correct as far as terms in x 3 ; where y~ 2 — (4a) -2 + £ 2 . 

This integral is to be taken throughout the extent of the 
wave-front. The phenomena of optics show however that it is 
only the parts of the wave-front in the neighbourhood of the 
normal that are efficient in producing illumination along the 
normal, for the more remote parts may be blocked out without 
affecting it. The integral may therefore be confined to the im- 
mediate neighbourhood of the origin, and we may proceed by 
approximation. Taking £ to be small of the same order as b, we 
have as far as cubes 

r = 7 _1 — y%x — ^yba~ l x 3 , 
ds=dx(l + 2aV + §abx s ), 
and i is of the form i = i (1 + ax + fix 2 + yx 3 ), 
t being the amplitude at the origin. 

J V t \y A, a\ J l 

Writing x' = x + \ax 2 + £ (2a 2 + /3) a fi , 

so that x = x'-\ ax' 2 - i (2a 2 + /3) x' 3 + } aV 3 , 

) ( 27rf _ 2tt + 2^y| ^ _ tttoI ^ 

( T X7 X \ 

+ i [- %iry%{2a 2 -f/3) + 7T 7 £cr + tt 7 6] a*"} aV, 

A, J 

12—2 






this becomes 



134 Mr Larmor, On the Laws of the [Jan. 26, 

or say 



which 



( [sin j 2 ^ (t - ~) + Act + Bx 2 + cA dx\ 
= lJ)~ 1 (sin ]-—(* — const.) + \k (w a — mw) dw, 



where w = D ( x + ^ n J , so that £77 D 3 = (7, 



3(7, 

and - \irm = i ^ - ^J = (£ir)* 40-* (l - ^J ; 
which gives on reduction, writing unity for j/2a, 

- m = 6"* (|j* 2af |l - £ (K - &a» " 1/3) I j , 

so that 

2a£ = m (i\f b h {1 + \rno-* (\\f (\d l - ^a 2 - $0) £}, 
or in terms of the radius of curvature (R = l/2a) of the wave-front 
at the origin and the radius of curvature p of the caustic 

3. Neglecting the term of the second order in this expression 
we have £ = — m ( ~ J , 

showing that the course of the ray caustic is bordered by a series 
of fringes which remain similar to each other throughout, and 
therefore are of the same type as the asymptotic fringes of Airy's 
supernumerarjr rainbow; but they come closest together at places 
of greatest curvature of the caustic according to the law that 
their separation at any place is proportional to the cube root of 
the radius of curvature of the caustic at that place. 

The investigation shows that unless for fringes at a considerable 
distance from the ray-caustic their form is not sensibly affected by 
the varying intensity in the wave-front. 

As we proceed along a caustic, the curvature gradually in- 
creases and the fringes therefore come together when we approach 
a cusp. At the cusp itself 6 = 0; and very near to it b is very 
small, so that to determine the state of matters for an unlimited 
beam another term would have to be included in the equation 
of its front; but if the beam is limited in any way the fringes 
produced by this limitation will there rise in importance, and 
practically obliterate the ones now under discussion 1 . 

1 See Rayleigh, "Investigations in Optics," Phil. Mag., Nov. 1889, pp. 408—10. . 



1891.] Diffraction at Caustic Surfaces. 135 

4. The results of this analysis indicate how we may proceed 
in the general case where the wave is not cylindrical, but is curved 
in two dimensions. 

Referred to the normal as axis of z, and the tangents to the 
arcs of principal curvature as axes of x and y, the equation of its 
front is 

z = ax* + bf + px* + Sqx 2 y + 3rxy* + sif 4- ... 

It is required to find the disturbance propagated to the point 

1 



6 0,. 



m 



Here 



-(r+e-sf+'+v-i 

= y" - 7 jf* + (i - £) y" - v* - H«?y - 3» V - «/} , 

so that I It sin ( — -^jdxdy 

will be complicated. 

But the considerations already mentioned show that the value 
of the integral is practically settled by the elements in the neigh- 
bourhood of the origin, for which x and y are small. We may 
therefore consider only a small rectangular portion of the wave- 
front bounded by arcs parallel to the axes of x and y. To deter- 
mine the diffraction in the plane z = l/2a, we may consider only 
the plane problem presented by a wave of the form 

z = ax 1 + px 3 ; 

for the uniform curvature in the perpendicular plane represented 
by the coefficient b will not affect the result at all, as is also 
obvious on continuing the general calculation. The variation of 
that curvature represented by the coefficient r will slightly dis- 
place the fringes, as it will alter the mean value of y%. 

The dissymmetry indicated by the coefficients q and s will on 
the average produce no effect on the disturbance at a point in the 
plane xz; these coefficients introduce odd powers of y which 
integrated over equal positive and negative range leave no 
appreciable result. 

Thus the illumination in the plane z= l/2a is determined by 
the values of a and p only. 



136 Mr Larmor, On the Laws of the [Jan. 26, 

Every beam in a homogeneous medium therefore converges to 
two ray-caustic surfaces which are the two sheets of the surface 
of centres of curvature of the wave-fronts. Each of these surfaces 
is physically made up of a series of parallel bright and dark sheets, 
of which the first is much the brightest, whose distances and 
relative intensities always retain the same proportions. These 
distances are at any point proportional to the cube root of the 
radius of curvature of the normal section of the caustic surface 
containing the ray which touches it at that point. 

5. It is easy enough to obtain actual examples of this general 
proposition. On looking at a bright lamp, sufficiently distant 
to be treated as a luminous point, through a plate of glass covered 
with fine rain-drops, the caustic surfaces after refraction through 
the drops are produced within the eye itself, and their sections by 
the plane of the retina appear as bright curves projected into the 
field of vision. These curves are each accompanied by the other 
parallel diffraction bands, which separate and become more marked 
as the curves recede asymptotically, while they assume a dif- 
ferent character near the cusps which are a feature of all sections 
of caustic surfaces. Near these cusps in the cross-section of the 
caustic surface two different pencils of light come into inter- 
ference. 

These phenomena are quite different from the ordinary cases 
of entoptic diffraction, in which when the eye is put out of focus 
by a lens, and a bright sky is viewed through a pinhole, the 
pencil of light coming through the pinhole projects on the 
retina shadows of the muscae volitantes floating in the aqueous 
humour, and these are accompanied by the ordinary bands at the 
boundaries of shadows. This case of an obstacle is the exact 
complement of that of a similar hole in a screen as regards the 
position of the bands ; so that when the obstacle is small, the 
diffraction bands round the shadow form exact circles, irrespective 
of its shape, which is the ordinary visual appearance. 

The cusped caustic bands are easily seen when a distant street- 
lamp is viewed through a spectacle lens with minute rain-drops 
deposited on it. 

The diffraction problem which has here been discussed includes 
diffraction at a focal line, with an unlimited beam. In practical 
questions such as those relating to spectroscopes and the Her- 
schelian telescope, the exact focussing would however introduce 
the nature of the aperture into the discussion, and the limits of 
the integral would enter. 

6. In the case of a cylindrical beam the bands near the caustic 
have been counted up to 30 or more by W. H. Miller, and the 
divergence of the more remote ones is too great to allow an 



1891.] Diffraction at Caustic Surfaces: 137 

approximate theory like the above to be applied with much 
certainty. For them, as Prof. Stokes has remarked, a perfectly 
satisfactory procedure is to simply consider the difference of path 
of the two pencils of light which reach a given band by different 
ways ; these may be considered as two separate interfering rays, 
exactly in the manner of Thomas Young's first apergu of the super- 
numerary rainbow. Now the difference of paths of the two rays 
up to the point P is clearly the excess of the two tangents from 
P to the geometrical caustic over the arc between their points of 
contact. Thus we obtain the following simple and elegant graphical 
construction, which applies to all the system of bands except the 
first two or three ; imagine the caustic curve constructed as a 
disc, and let an endless thread be placed round it, the bands will 
be traced by a pencil strained by this thread in the same manner 
as in the ordinary construction of an ellipse by a thread passing 
round its foci ; and successive bands will correspond to equal 
increments in the length of the thread. 



(3) The effect of Temperature on the Conductivity of Solutions 
of Sulphuric Acid. (Plates IV. and V.) By Miss H. G. Klaassen, 
Newnham College (communicated by Prof. J. J. Thomson). 

Graham has shown that the viscosity of sulphuric acid increases 
upon addition of water until a maximum is reached at the com- 
position of nearly one molecule of water to one of acid. Upon 
further dilution the viscosity continually decreases. 

If a curve, in which the ordinates represent the electrical 
resistance and the abscissae the percentage composition of solutions 
of sulphuric acid in water, be drawn from Kohlrausch's observa- 
tions, it will be seen that a point of maximum resistance exactly 
corresponds to this degree of concentration. The fact, that these 
maxima should both occur when the composition of the solution 
is almost exactly one molecule of water to one of acid, points to the 
existence of the hydrate H 2 S0 4 . H 2 0. 

If the increase of viscosity and of resistance, supposing it due 
to this hydrate, were found to diminish with a rise of temperature, 
this fact would furnish strong evidence in favour of the theory 
that the hydrate H 2 S0 4 . H 2 dissociates into H 2 S0 4 and H 2 at 
higher temperatures. 

That the increased viscosity does diminish with the rise of 
temperature has been proved by some recent determinations of 
the viscosity of sulphuric acid solutions at temperatures between 
15° C. and 100° C. 1 

1 Phil. Mag., Oct. 1N89. 



138 



Miss Klaassen, On the effect of Temperature on [Jan. 26, 



It was suggested to me by Professor Thomson that it would be 
of interest to ascertain if the increased electrical resistance due to 
the hydrate would also tend to disappear at higher temperatures. 

With this object in view I determined the resistance of various 
solutions of sulphuric acid at temperatures between 15° C. and 
100° C. 

The resistance was measured by means of the Wheatstone 
bridge with a Post Office resistance box and a double com- 
mutator 1 which reversed the battery and galvanometer circuits 
simultaneously. In every experiment the solution was heated 
twice to a temperature of nearly 100° C, but in no case did the 
observations taken indicate any change of concentration from 
absorption of steam. 

The following observations were taken : 



97 °/ H„SO, 



Temperature Besistance 

17-4° C 27-4 Ohms 

97-6 6-8 

86-6 7-7 

79-2 8-4 

70-9 9-5 

59-0 Ill 

48-2 13-5 

37-9 16-5 

29-5 20-0 

17-0 27-9 



Temperature Besistance 

86-5° C 7-7 Ohms 

85-5 7-8 

80-4 8-3 

71-2 9-4 

591 11-2 

46-5 14-0 

427 150 

35-6 17-5 

302 19-8 



95% H 2 SO, 



Temperature Besistance 

16-2° C 20-3 Ohms 

90-9 5-37 

87-0 5-6 

78-7 6-22 

70-5 6-9 

61-0 7-9 

49-5 9-6 

39-6 11-7 



Temperature Besistance 

64-0° C 7-6 Ohms 

48-2 9-8 

34-1 13-0 

206 18-1 

18-9 18-85 

90-0 5-4 

94-5 5-1 

81-1 61 

67-0 7-3 



1 For description of commutator, see Brit. Ass. Report, 1886, p. 328. 



1891.] the Conductivity of Solutions of Sulphuric Acid. 139 



90-4°/ o 

Temperature Eesistance 

16-4° C 19-22 Ohms 

81-4 4-9 

79-9 5-0 

67*3 6-1 

92-25 42 

90-6 4-3 

70-9 5-22 

66-9 61 

50-5 8-3 

37-5 10-9 

39-0 10-5 

67-4 6-06 

86-5% 

Temperature Resistance 

14-7° C 22-2 Ohms 

34-6 1218 

66-9 61 

64-8 6-3 

440 9-6 

35-1 12-0 

16-2 211 

84-1 °/ 

Temperature Eesistance 

15-7° 21-62 Ohms 

24-65 16-51 

35-05 11-9 

43-4 9-7 

54-2 7-7 

62-8 6-4 

81-7 °/ 

Temperature Resistance 

17-3° C 19-85 Ohms 

67-7 5-7 

59-5 6-6 

47-0 8-6 

29-9 13-3 

87-2 4-1 

92-0 3-83 

81-9 4-5 

80-5 4-6 



H 2 S0 4 

Temperature Eesistance 

58-0° C 7-1 Ohms 

49-2 8-4 

47-7 8-7 

37-1 110 

29-1 13-4 

17-1 18-82 

89-2 4-4 

75-6 5-3 

50-7 8-2 

28-9 13-5 

14-2 205 

24-7 150 

H 2 S0 4 

Temperature Eesistance 

94-2° C 4-0 Ohms 

92-4 4-1 

85-2 4-5 

72-5 5-5 

61-9 6-6 

47-9 8-8 

23-45 16-7 

H 2 S0 4 

Temperature Eesistance 

68-5° C 5-8 Ohms 

86-9 4-4 

16-6 211 

38-7 10-95 

66-4 6-05 

54-8 7-6 

H 2 S0 4 

Temperature Eesistance 

16-3° C 20-3 Ohms 

57-2 0-9 

39-1 10-4 

37-5 10-8 

15-9 206 

87-2 4-1 

80-5 4-6 



140 



Miss Klaassen, On the effect of Temperature on [Jan. 26, 



74-7 % 

Temperature Eesistance 

16-8° C 13-3 Ohms 



96-7 
93-6 
82-5 
80-5 
68-9 
50-3 
42-0 
30-5 
28-7 



3-2 
3-3 

3-8 
3-9 
4-6 
6-2 
7-3 
9-4 
9-8 



H 2 S0 4 

Temperature 

91-9° C. 
87-9 


Resistance 

3-35 Ohms 
.... 35 


81-8 




.... 3-8 


71-4 




.... 4-4 


50-2 




.... 6-2 


42-0 




.... 73 


30*5 




.... 9-4 


28-0 




.... 100 


60-4 




.... 5-2 


63-2 




... 4-97 



30% 

Temperature Eesistance 

170° C 2-68 Ohms 



420 
624 
91-3 

84-0 



1-89 
1-60 
1-33 
1-40 



HS0 4 

2 4 

Temperature Eesistance 

63-6° C 1-53 Ohras 

46-0 1-81 

15-8 2-7 

•6 3-67 



These observations are represented graphically in Plate IV. in 
which Temperatures are represented by ordinates, and Resistances 
by abscissae. 

The form of these curves is similar in type to that of the 
corresponding ones for viscosity given by MLr D'Arcy in the Phil. 
Mag., Oct. 1889. The curvature first increases with the tempera- 
ture, reaches a maximum, and then decreases. 

In Plate V. are represented the isothermals for 18° C, 20° C, 
30° C, 40° C, 50° C, 00° Q, 70° C, 80° G. and 90° C, drawn from 
the curves in Plate III., resistances being represented by ordinates, 
and the percentages of H 2 S0 4 by abscissas. Points marked are 
taken from Kohlrausch's observations. 

The isothermal for 18° C. has been continued up to its minimum 
point (30% H 2 S0 4 ) from the observations of F. Kohlrausch. 

This curve has a point of minimum resistance at about 
92 % H 2 S0 4 , 
the resistance then rises, reaching a maximum at 

84-5 % H 2 S0 4 (the hydrate H 2 S0 4 . H 2 0), 
and then again falls. The isothermals for higher temperatures 
show that this rise in the resistance gradually diminishes, up to 
70° C, above which temperature the resistances are in descending 
order of magnitude. The increased resistance due to the hydrate 
has, however, not quite disappeared, for even in the isothermal for 
90° C. there is a change in the sign of the curvature, although 
there is no longer a point of maximum resistance. 



1891.] the Conductivity of Solutions of Sal pit uric Acid. 141 

Additional evidence in favour of the existence of the hydrate 
H 2 S0 4 . H 2 is afforded by the fact that it can be obtained in the 
crystalline form. 

The hydrates of S0 3 which have at present been crystallised 
are: 

melting point 

(1) H 2 S 2 7 

(2) H 2 S0 4 0°C. 

(3) H 2 SG 4 .H 2 7-5° C. 

(4) H 2 S0 4 .4H 2 -25° C. (Pickering, Chem. Netos, 

1889). 

We know from the experiments of W. Kohlrausch upon solu- 
tions of S0 3 in H 2 S0 4 , and from those of F. Kohlrausch upon 
aqueous solutions of H 2 S0 4 , that the formation 1 of (2) and (3) is 
accompanied by an increased electrical resistance. H 2 S0 4 . 4H 2 0, 
on the other hand, appears to have no effect upon the resistance at 
ordinary temperatures. What effect the formation of the hydrate 
H 2 S 2 7 may have upon the resistance has not been ascertained, as 
the observations of W. Kohlrausch do not extend to solutions of 
greater concentration than 90'67 °/ S0 3 . 

Judging from these facts it would seem that the formation of 
a hydrate does not necessarily produce an increased resistance, 
though in the cases of H 2 S0 4 and H 2 S0 4 . H 2 it appears to do so. 

But we have seen that the hydrate H 2 S0 4 . H 2 dissociates as 
the temperature rises. It is possible, therefore, that H 2 S0 4 . 4H 2 
which has a melting point of — 25° C, while that of H 2 S0 4 . H 2 is 
7'5° C, may be so far dissociated at 18° C. that there is not 
sufficient of the hydrate present to produce a perceptible effect 
upon the resistance. 

February 9, 1891. 

Professor G. H. Darwin, President, in the Chair. 

The following Communications were made to the Society : 

(1) On Rectipetality and on a modification of the Klinostat. 
By Dorothea F. M. Pertz and Francis Darwin, M.A., Christ's 
College. 

[Abstract; received March 5, 1891.] 

Vochting has shown that geotropically induced curvatures 
may be removed by subjecting the curved plant to slow rotation 
on a horizontal klinostat. It is usually assumed that the growing 
part, being freed from external stimulation, straightens itself by 

1 The point of maximum resistance is between 99"75 / o and 99*9 °/ H 2 S0 4 . 



142 Mr Shipley, On the Occurrence of Bipalium [Feb. 9, 

an inherent regulating power to which Vochting has given the 
name Rectipetality. But it is not certain that the klinostat does 
remove external stimuli. Elfving's experiments on the growth of 
grass-halms show that though the gravitation-stimulus is sym- 
metrically distributed, it is not destroyed. If the klinostat's 
action depends on the symmetrical distribution of stimuli, not on 
their removal, we shall be compelled to take a different view of rec- 
tipetality. To test this question a new form of horizontal klinostat 
was devised. The axis of this instrument is not kept in constant 
rotation, but at intervals of half-an hour it executes a half revo- 
lution. A shoot or stem geotropically curved is fixed in the 
klinostat so that the plane of curvature is horizontal. The suc- 
cession of half turns prevents any geotropic distortion in the plane 
at right angles to the original plane of curvature, while in that 
plane the plant is free to increase or diminish its curvature apart 
from any fresh gravitation-stimulus. The experiments show that 
under these circumstances the curvature diminishes, and this can 
only be due to an inherent regulating power, the rectipetality of 
Vochting. 

The modified klinostat, which will probably be of use for 
other purposes, was designed and made by the Cambridge Scien- 
tific Instrument Company. 

(2) On the Occurrence of Bipalium Kewense, Moseley, in a 
neiu Locality; with a Note upon the Urticating Organs. By 
Arthur E. Shipley, M.A., Christ's College. 

[Received February 9, 1891.] 

Bipalium Kewense was first described by Professor Moseley in 
the year 1878, from specimens obtained in the hot-houses at 
Kew. In 1883 Dr Giinther received some examples of this species 
from Welbeck Abbey, and one specimen was described from Clap- 
ham Park. Mr O. Salvin exhibited some of these animals before 
the Zoological Society in 1886, collected from amongst pieces of 
broken tiles at the bottom of some pots of Calceolarias which had 
stood in a cold frame all the winter in his garden near Haslemere, 
Surrey. Some specimens from the same locality formed the object 
of some interesting observations by Professor Jeffrey Bell during 
the same year 1 . Finally the specimens which I am able, owing to 
the kindness of Professor Newton, to exhibit to the Society this 
evening came from the neighbourhood of Bath. 

1 Since writing the above, Prof. Herdman has informed me that Bipalium 
Kewense was found in an Orchid house at Aigburth, near Liverpool in 1888, 
and Mr Beddard writes to me that specimens are from time to time brought to him 
from the gardening department of the Zoological Gardens. 



1891.] Kewense, Museley, in a new Locality. 143 

Outside England this species has been found in the Orchid- 
Houses of the Botanical Garden in Berlin, and in the Palm- 
Gardens of Frankfort. As early as 1883 it had been noticed in 
the Botanic Garden of Cape Town, and it has been known in 
Sydney since 1874, although its presence was not registered till 
1888. It has recently been found by Mr J. J. Lister under stones 
in the forests of Upolu, Samoa, and it is not impossible that this 
is one of its native habitats 1 . 

Unfortunately the records of the appearance of B. Kewense 
fail to throw much light on its native habitat. It has almost in- 
variably turned up in hot-houses, usually associated with Orchids ; 
and there is nothing to show that these Orchids in all cases came 
from the same part of the globe. Sydney is the only place where 
it seems to have established itself, and here it is described as 
existing in great numbers, lying under pieces of wood, etc., or 
crawling along the pavements and palings. In England, as is 
pointed out above, it has only occurred sporadically and a few at 
a time, probably introduced afresh, or disseminated from Kew. 
It does not seem in any danger of establishing itself. Rhyncho- 
demus terrestris, 0. F. Mull, still remains the only British Land 
Planarian. 

The specimens from Bath vary a good deal in length. One, 
whilst crawling up a wall, attained the length of seven inches. 
The body is extremely extensile and soft, and the animals seem 
capable of creeping through the smallest crannies. Those that 
Professor Moseley described were some of them nine inches long, 
a length surpassing that of any other species of Bipalium, but 
New South Wales specimens were even longer ; one of them 
measured 14 in. and another 9 in. (6). The worms crawl about 
actively, by means of the strong cilia on their ventral surface, and 
in their natural conditions appear to coil round stems and blades of 
grass, etc. 

The semilunar anterior end, which is characteristic of the 
genus, does not always maintain its outline, but the contour of the 
head is constantly changing (1). The head is generally raised 
above the surface of the ground, and processes appear to be 
pushed out from its edge, whicli test the surface upon which the 
animal crawls. 

There seems to be no doubt that B. Keiuense, like other allied 
species, is nocturnal in its habits, strong sunlight being harmful 
and often fatal to it. Its surroundings must also be kept very 
damp. One of the most curious features of its economy is the 
harmful effect which certain foreign substances have upon its well- 
being. Dr Trimen (27) records how one specimen was killed in 

1 Vide Zoolog. Anzeiger, No. 361, p. 139. 



144 Mr Shipley, On the Occurrence of Bipalium [Feb. 9, 

the space of some minutes by dropping upon a grate covered with 
blacking ; Mr Fletcher (6) states that in Sydney, where they ap- 
peared in considerable numbers, they were constantly found either 
moribund or dead on the pavement, the surface of which ap- 
parently did not agree with them, and the living specimens which 
were kindly sent to Cambridge by Mr L. Birch unfortunately 
crawled on to the sides of a glass bottle and there died. Mr 
Harmer tells me that the same fate overtook some specimens of 
Rhynchodemus terrestris which had found their way on to a glass 
surface. This extreme sensibility to the contact of foreign sub- 
stances seems strange when we remember what a copious coating 
of mucus these animals can produce at will. 

Bipalium Kewense lives on earthworms, insects, etc. and probably, 
like the other species of the same genus whose habits have been 
investigated, it is entirely carnivorous, and does little harm to the 
orchids and other plants with which it may be associated. Some 
species devour small molluscs as well as earthworms, the radula of 
a snail having been found within a Bipalium from the Philippines 
(12). B. javanum (14) eats small Gastropods; its pharynx, which 
envelopes the body of the animal, shell and all, being powerful 
enough to crush the shell, the pieces of which do not pass into the 
alimentary canal, and are either rejected or possibly dissolved by 
the secretion of the numerous glands which open into the pha- 
rynx. At present there is no evidence to show that these animals 
are ever vegetable feeders. 

Like most other Turbellarians, Bipalium is hermaphrodite ; 
but it also reproduces by transverse fission, and this may to some 
extent account for the great variation in length in individuals 
found in the same locality, and also for the not unfrequent absence 
of the semilunar head. B. javanum is protandrous, the male 
organs being mature in July and August, the egg-cocoons not 
being deposited till October or November (14). 

The faculty of depositing considerable quantities of mucus is 
one of the most remarkable characteristics of these animals ; their 
path can be traced by a slimy tract which quickly dries up. Prof. 
Jeffrey Bell's (1) observations tend to show that in Bipalium 
Kewense the mucus is secreted by the anterior end of the body : 
this observer is of opinion that the secretion may serve to entangle 
offending bodies, and possibly also helps to catch objects which serve 
as food. In some of the tropical species this mucus is very 
copious, and hardens into threads by means of which the animals 
suspend themselves. They are occasionally blown, hanging at 
the end of their threads, from one stem or branch to another, like 
the young of many species of spiders. 

The specimens which were sent to Cambridge had unfor- 
tunately crawled on to the inner surface of the glass bottle in 



1891.] Kewense, Moseley, in a new Locality. 145 

which they were confined. When they came into my hands they 
were in a state of deliquescence, and formed a slimy mass with 
a more or less definite outline. On examining this deposit with 
a microscope, it became apparent that the various cells composing 
the body of the animal had parted company. Amongst the 
numerous glandular and other cells which formed a considerable 
portion of the slime, certain large cells were seen, each of which 
appeared to contain two or more bodies. At first I mistook these 
structures for the ordinary rod-shaped rhabdites, so common in 
Planarians. On a more thorough examination, however, I found 
that what I had at first taken for two bodies was in fact but one 
continuous structure bent in the form of a V; and that to one end 
of the body a long whip-like appendage was fixed. Several of the 
cells, which were provided with a nucleus, burst while under ob- 
servation ; and they appeared to do so by the efforts of the con- 
tained rods to straighten themselves ; at any rate, when the cell 
burst, the rods straightened themselves, and the thread borne by 
their ends was thrown out. The thread did not, like the threads 
of the nematocysts of Hydra, stretch straight out, but assumed 
a somewhat coiled disposition. The rhabdites, which were also 
present, had very much the appearance of the thicker basal portion 
of the flagellated structures, but were without the whip ; and it 
occurred to me that possibly the rhabdites, which are so common 
in the Turbellaria, might be derived from the basal part of the 
flagellated structures. This view is, to a slight extent, supported 
by the fact observed by Loman that, in the East Indian species he 
examined, the number of the simple rhabdites and the number of 
flagellated structures varied inversely. Loman is inclined to 
regard the somewhat similar bodies, prolonged at each end into 
a fine thread, which he found in his Bipalium javanum, as true 
nematocysts, such as are found in Coelenterates and in some 
species of Rhabdocoels, e.g. in Microstoma lineare and Stenostoma 
Sieboldii. But whereas in the typical nematocyst the urticating 
thread is coiled up inside the capsule, and is evaginated when 
shot out, in the bodies found in Bipalium there is no capsule, but 
a basal thick portion, either bent or coiled, and a thin thread 
wound round this. That these organs have, at any rate in some 
species, the same irritating properties as the nematocysts of Coe- 
lenterates is shown by the fact that Mr Thwaites (IS) experienced, 
when he applied his tongue to some living land Planarians in 
Ceylon, " a feeling of unpleasant tingling," which " was accom- 
panied with slight swelling. The sensations [were] very similar 
to what is experienced upon a slight scalding." Mr Dendy also 
tasted the Australian species, Geoplana Spenceri, and describes the 
results as very unpleasant. 

The following is a list of the more important papers which 



146 Mr Shipley, On the Occurrence of Bipalium [Feb. 9, 

have appeared on Land Planarians since the publication of Pro- 
fessor Moseley's memoir (17). 

1. Bell, F. Jeffrey; 

Note on Bipalium Kewense, and the Generic Characters of Land-Planarians. 
Proc. Zool. Soc, 1886, p. 166. 

2. Beegendal, D.; 

Zur Kenntniss der Landplanarien. 

Zool. Anzeiger, x. Jahrgang, 1887, p. 218. 
— Do. Ann. and Mag. Nat. Hist. (5th Ser.), Vol. 20, p. 44. 

3. Blomefield, L. ; 

Note on the Occurrence of the Land Planaria (Planaria terrestris) in the 
neighbourhood of Bath. 
Proc. Bath Nat. Hist, and Antiquarian Field Club, Vol. in., 1887, p. 72. 

4. Carriere, J.; 

Ein neuer Fundort von Planaria terrestris, O. F. Muller. 
Zool. Anzeiger, n. Jahrgang, 1879, p. 668. 

5. Dendy, Aethue; 

Anatomy of an Australian Land Planarian. 
Trans. Roy. Soc. Vict., Vol. i. Pt. n., p. 50. 

6. Fletcher, J. J.; 

Remarks on an introduced species of Land Planarian, apparently Bipalium 
Kewense, Moseley. 
Proc. of the Linnean Soc. of New South Wales, 2nd Ser., Vol. n., 1887, p. 244. 

7. Fletchee, J. J., and Hamilton, A. G. ; 

Notes on Australian Land Planarians with Descriptions of some New Species. 
Proc. of the Linnean Soc. of New South Wales, 2nd Ser., Vol. n.,p. 349. 

8. v. Graff, L.; 

Ueber ein. interessante Thiere des Zoolog. und Palmengartens zu Frankfurt a. M. 
Der Zoolog. Garten, 20 Jahrgang, 1879, p. 196. 

9. Harmer, S. F.; 

[Note on the occurrence of Rhynchodemus terrestris at Cambridge.] 
Proc. of the Camb. Philos. Soc, Vol. vn., Pt. n., p. 83. 

10. v. Kennel, J. ; 

Die in Deutschland gefundenen Land-Planarien, Bhyncodemus terrestris, 
0. F. Muller, und Geodesmus bilineatus, Mecznikoff. 
Arbeiten aus dem Zoologisch-Zootomischen Institut in Wiirzburg, Bel. v., 1882, 
p. 120. 

11. . — Bemerkungen iiber einheimische Land-Planarien. 

Zool. Anzeiger, i. Jahrgang, 1878, p. 26. 

12. Loman, J. C. C; 

Ueber den Bau von Bipalium, Stimpson. 

Bijdragen tot de Dierhunde, Aflev. 14, p. 61, 1887. 

13. — Zwei neue Arten von Bipalium. 

Zool. Anzeiger, Vol. vi., 1883, p. 168. 

14. — Over den bouw van de Land-Planarien. 

Tijdschr. Nederl. Dierk. Ver., i. Deel, p. 130. 

15. — Ueber neue Landplanarien von den Sunda-Inseln. 

Zool. Ergebn. ein. Reise in Niederl. Ost. Ind. Heft. i. p. 131. 

16. Moseley, H. N. ; 

Description of a new Species of Land-Planarian from the Hot-houses at Kew 

(~r fl Y cl 6 Tl S 

Ann. and Mag. of Nat. Hist., 5th Ser., Vol. i., 1878, p. 237. 



1801.] Kewense, Moseley, in a new Locality. 147 

17. Moseley, H. N.; 

On the Anatomy and Histology of the Land Planarians of Ceylon, with some 
Account of their Habits and a Description of two new Species, and with 
Notes on the Anatomy of some European Aquatic Species. 
Phil. Trans., 1873, p. 105. 

18. — ; 

Urticating Organs of Planarian Worms. 
Nature, Vol. xvi., 1877, p. 475. 

19. — Notes on the Structure of several Forms of Land Planarians, etc. 

Quart. Journ. Microsc. Sci., Vol. xvn., 1877, p. 273. 

20. de Man, J. G.; 

Geocentrophora sphyrocephala n. gen. n. sp., eene landbewonende Bhabdoeoele. 
Tijdschr. Nederl. Dierk. Ver., n. Deel, 1875. 

21. — De gewone europeesche Land-planarie, Geodesmus terrestris 0. F. Mull. 

Tijdschr. Nederl. Dierk. Ver., n. Deel, 1876, p. 238. 

22. Eichtees, Feed.; 

Bipalium Kewense eine Landplanarie des Palmenhauses zu Frankfurt a. M. 
Der Zoolog. Garten, xxvm. Jahrgang, 1887, p. 23. 

23. Salvin, 0.; 

Exhibition of Bipalium Kewense. 
Proc. Zool. Soc, 1886, p. 205. 

24. Schulze, F. E.; 
Uber lebende Bipalium. 

Sitzungsber. Ges. Nat. Freunde Berlin, 1886, p. 159. 

25. Spencer, W. B. ; 

Notes on some Victorian Land Planarians. 
Proc. Royal Soc. of Vict., 1891, p. 84. 

26. Steensteup, J.; 

Om Jord-Fladormens (Planaria terrestris 0. F. M.) Forekomst i Danmark. 
Vid. Medd. f. d. naturh. Forening i Kjobenhavn, 1869, p. 189. 

27. Teimen, Poland; 

On Bipalium Kewense at the Cape. 
Proc. Zool. Soc, 1887, p. 548. 

28. Vejdovsky, F.; 

Note sur une nouvelle Planaire terrestre (Microplana humicola nov. gen., 
nov. sp.). 
Rev. Biol. Nord de la France, n., 1889-90, p. 129. 

29. Zachaeias, 0.; 
Landplanarien auf Pilzen. 

Biolog. Centralblatt, 8 Bd., 1888-1889, p. 542. 

(3) The Meclusce of Millepora and their relations to the niedu- 
siform gonophores of the Hydromedusse. By S. J. Hickson, M.A., 
Downing College. 

[Abstract; reprinted from the Cambridge University Reporter, Feb. 17, 1891.] 

In Millepora plicata no medusiform structures of any kind 
were observed. The spermaria are simple sporosacs on the sides 
of the dactylozooids. The eggs are extremely minute and show 
frequently amceboid processes : they are found irregularly dis- 

VOL. VII. PT. IV. 13 



148 Mr MacBride, On the Development [Feb. 9, 

tributed in the coenosarcal canals of the growing edges of the 
colony. 

In Millepora murrayi from Torres Straits large well-marked 
medusae, bearing the spermaria, were observed lying in ampullae 
of the ccenosteum. Even when free from the coenosarcal canals 
and ready to escape they show no tentacles, sensory bodies, radial 
or circular canals, velum or mouth. They are formed by a simple 
metamorphosis of a zooid of the colony. The eggs of this species, 
like those of M. plicata, are extremely small and amoeboid in 
shape. They are not borne by special gonophores. 

In the Stylasteridce, the eggs are large, contain a large quantity 
of yolk, and are borne by definite cup-like structures produced by 
foldings of the coenosarcal canals. 

In Allopora the spermarium is enclosed by a simple two 
layered sac composed of ectoderm and endoderm. The endoderm 
at the base is produced into the centre of the spermarium as 
a simple spadix. 

In Distichopora the male gonophores are similar to those of 
Allopora, but there is no centrally placed endodermal spadix. In 
both genera a two layered tube (seminal duct) is produced at the 
periphery of the gonophore when the spermatoza are ripe. 

Neither the gonophores of Allopora and Distichopora, nor the 
medusae of Millepora murrayi show any traces in development of 
being degenerate structures like the adelocodonic gonophores of 
the other Hydromedusce. 

(4) The Development of the Oviduct in the Frog. By E. W. 
MacBride, St John's College. 

[Abstract: received February 9, 1891.] 

In July of last year I undertook, at Mr Sedgwick's suggestion, 
the investigation of the origin and growth of the oviduct in the 
Frog. Some considerable time was spent in determining the 
stages in which it is formed. No trace of it is visible till the 
animal has lost all the characteristic tadpole organs except the 
tail; and it is complete, from a morphological point of view, 
when the frog has attained a length of about 17 millimetres, after 
the tail has been absorbed. 

So far as I am aware, there are only two papers dealing with 
the development of this duct in the Anura. The first of these is 
by C. K. Hoffmann, in the 'Zeitschrift fur wissenschaftliche Zoo- 
logie ' for 1886 (Bd. 44), and the other' by A. M. Marshall and 
E. J. Bles, in the second volume of the ' Studies from the Biolo- 
gical Laboratories of the Owens College.' 

Hoffmann describes the duct as arising from a patch of modified 
peritoneum just ventral to the third and now only remaining 



1891.] of the Oviduct in the Frog. 149 

nephrostome of the pronephros. This patch, he says, is dorsally 
involuted to form a groove, open below. Ventrally it is prolonged 
downwards and outwards over the surface of the pronephros, and 
even beyond it for a short distance. It is distinguished from the 
unmodified epithelium by being highly columnar. The part of 
the Wolffian duct in front of the mesonephros, the lumen of which 
is at this stage much reduced, then separates itself from the de- 
generating pronephros, and splits into two rods of cells. The 
dorsal of these is continuous with the Wolffian duct behind, and 
the ventral one applies itself in front to this involuted patch of 
peritoneum, and forms the first rudiment of the oviduct. But the 
oviduct does not grow back in continuity with the Wolffian duct. 
On the contrary, it enters into close connection with a longitudinal 
strip of peritoneum which lies to the outer border of the kidney, 
and consists of columnar epithelium. Hoffmann states that the 
hinder portion of the duct is formed from these cells, though 
whether by involution to form a groove or by proliferation he 
could not determine. In the meantime, the groove which formed 
the ostium of the duct, and which was originally dorsal, has 
become prolonged ventrally round the base of the lung. It closes, 
forming a canal which now opens ventrally. Later, this ventral 
extension atrophies, leaving the ostium in its original dorsal 
position. 

Marshall and Bles have only observed a few isolated facts in 
the development of the oviduct. They confirm Hoffmann in his 
account of the ventral displacement of the ostium ; but fail to 
observe any splitting of the front part of the Wolffian duct. They 
state that the hind end of the duct is, in the first year, a solid rod 
of cells ; but do not notice any relation to the peritoneum. 

My observations differ from those of Hoffmann in several im- 
portant particulars. It will be convenient to describe first the 
origin and fate of the abdominal opening, and then that of the 
rest of the duct. In a tadpole in which the hind-limbs alone are 
visible, I find three nephrostomes in the head-kidney, the cells of 
which bear long flagella pointing inwards, as Hoffmann has 
pointed out. The first of these is situated some way in front of 
the glomerulus, the second immediately in front of the attachment 
of the glomerulus, and the third immediately behind it. 

In my next stage ; that is to say in a frog 38 mm. long (in- 
cluding the tail, which was about 21 mm. long), one nephrostome 
only remains ; but, as this is situated some considerable distance 
in front of the glomerulus, it must be regarded as the first and not 
the third of the preceding stage. Immediately ventral to it 
there is a groove in the peritoneum, open below, the epithelial 
lining of which is very columnar and quite different in appearance 
to that of the nephrostome. This columnar epithelium is con- 

13—2 



150 Mr MacBride, On the Development [Feb. 9, 

tinued out over the surface of the pronephros arid beyond it, as 
Hoffmann has described. The groove, traced backwards for 15 
sections, becomes a canal, which after two sections ends in a solid 
thickening of the peritoneum. In succeeding stages the groove 
extends ventrally, as described by Hoffmann, and in the last stage 
it has become a canal, opening somewhat ventrally, the opening 
having already acquired the fimbriae of the adult orifice. But 
I have been quite unable, after examining a number of adults, to 
detect any important difference between the position of the ab- 
dominal opening in my last stage and that of the adult funnel. 
The latter is not situated dorsally but at the side of the lung. It 
is fimbriated, and these fimbriae are continued, lying in a groove, 
on to the mesentery connecting the stomach and liver, over the 
ventral surface of the lung. It is true that the length of the 
orifice appears to have extended- somewhat in a dorsal direction, 
but that is all. 

As to the origin and backward growth of the duct behind the 
funnel, I find that the mode of origin described by Hoffmann for 
its hinder portion holds for its entire length. It has been men- 
tioned above that in the earliest stage the peritoneal groove, 
traced backwards, ends in a slight thickening of the peritoneum. 
The next stage shows the same condition of things; but a good 
length of the groove has been converted into a canal. In the first 
stage the Wolffian duct is still discernible in the part of the body 
in front of the mesonophros, but it is reduced to a rod of pale 
degenerate cells. There is no indication of any splitting such as 
Hoffmann described ; and it appears a priori in the highest 
degree improbable that fresh development should take place from 
such an atrophied rudiment. I may mention that Hoffmann 
omits to give any figures illustrating this point. In frogs of this 
age (that is to say those in which the tail is about as long as the 
body), there is a distinct line of epithelium on the outer border of 
the kidney, reaching back to the cloaca, which is distinguished 
from the rest of the peritoneum by its more columnar character. 
In succeeding stages, all trace of the Wolffian duct in front of the 
kidney is gone ; and the thickening of the peritoneum mentioned 
above travels back along this line of modified epithelium. I have 
called it a peritoneal thickening because I believe it to be derived 
from the peritoneum. It appears in sections as a projecting 
nodule of deeply staining tissue, the outermost cells of which pass 
at the side into the ordinary peritoneum. Furthermore, although 
the rudiment in front of the kidney grows back with some regu- 
larity, behind the kidney it is formed long before one can trace it 
at the side of this organ. It nowhere comes in contact with the 
Wolffian duct. But it is not possible to speak with absolute cer- 
tainty as to the origin of the cells which compose this solid rudi- 



1891.] of the Oviduct in the Frog. 151 

ment, because there is, especially in front, some lymphoid tissue 
at the outer border of the kidney — in fact the whole pointed end 
of the mesonephros degenerates to a string of such tissue. 

The lumen of the duct appears first in front and then behind 
in the region of the kidney. In the latter position it appears here 
and there in patches. It is formed by the rearrangement of some 
of the cells in the rod in a stellate manner, sometimes one cell 
and sometimes two cells deep beneath the surface. 

The conclusion which seems to be suggested by these investi- 
gations is the complete independence of the oviduct from the 
Wolffian duct in the Anura. 

I have, in conclusion, to express my warmest thanks to Mr 
Sedgwick for his advice and assistance to me in this work. 

February 23, 1891. 

Prof. G. H. Darwin, President, in the Chair. 

The following Communications were made to the Society : 

(1) Tidal Prediction— a general account of the theory and 
■methods in use and the accuracy attained. By Prof. G. H. Darwin. 

Published in Nature, Vol. 43, p. 609. 

(2) On Quaternion Functions, with especial Reference to the 
Discussion of Laplace s Equation. By J. Brill, M.A., St John's 
College. 

1. The following communication is intended as a sequel to 
the one that I made to the Society at the end of last term. In 
that paper I showed how we might obtain analogues to the 
theorems connected with conjugate functions with the aid of 
four related solutions of Laplace's Equation obtainable from the 
solution of a quaternion differential equation of the first order. 
I now propose to obtain a form for the general integral of the said 
equation. 

2. On account of the non-commutative character of the 
symbols involved, quaternion functions are of a more complicated 
character than ordinary scalar functions, and for their full dis- 
cussion would require a notation and nomenclature of their own. 
We may, however, in the case of functions of a single quaternion, 
as was done by Hamilton in the case of the exponential, extend 
the definitions of some of the ordinary scalar functions so as to 
apply to quaternions, by defining the quaternion function as the 
sum of a quaternion series exactly similar in form to the scalar 
series which defines the corresponding scalar function. It is to be 
remarked, however, that this method cannot be consistently carried 



152 Mr Brill, On Quaternion Functions, with especial [Feb. 23, 

out to the end, as the inverse of a quaternion function would not 
in general correspond with the quaternion function framed on the 
model of the inverse scalar function. Still further difficulties 
would arise if we attempted to apply the scalar notation to 
functions of two quaternions. 

So far as I am aware, Boole 1 was the first to give a general 
expression for a function of a single quaternion framed on the 
model of a specified scalar function. His expression may be easily 
deduced from Sylvester's Interpolation Formula 2 in the Theory of 

Matrices, which states that if \, \, \ be the latent roots of 

an w-ary matrix, then 

where m denotes the matrix, provided that none of the latent 
roots are equal. 

The identical equation satisfied by a quaternion is 

q*-2qSq+(Tqy = Q, 

and, therefore, the latent roots of the quaternion are given by the 
equation 

X i -2\Sq + (Tqf=0; 

and since (Tq) 2 = (Sq) 2 + (TVqf, 

it is clear that the roots are 

Sq + cTVq and Sq-iTVq. 

These roots are obviously distinct except in the case when q 
reduces to a scalar. Thus we have 

1 {f(Sq + iTVq)+f(Sq-LTVq)} 



2 



+ I UVq {/(Sq + cTVq) -f(Sq - cTVq)}, 

Ail/ 



which is Boole's result. 

1 " On the Solution of the Equation of Continuity of an Incompressible Fluid," 
Proc. E. I. A., vi. 375—385. 

2 This theorem was stated by Sylvester in a paper in the Phil. Mag. for 
October 1883, entitled " On the Equation to the Secular Inequalities in the 
Planetary Theory." It also appears as the second law in his "Laws of Motion 
in the World of Universal Algebra." 



1891.] Reference to the Discussion of Laplace s Equation. 153 

3. Assuming Boole's result arid writing f'(q) for the quaternion 
function framed on the model oif(x), we easily obtain 

V/(?) = Sf (q) . VSq - TVf (q) . VTVq + TVf{q) . V UVq 

+ [TVf (q) . VSq + Sf (q) . VTVq] . UVq 
= {VSq + VTVq . UVq] {Sf (q) + UVq . TVf(q)} 

+ TVf(q).VUVq 
= V? •/' (q) + VUVq. [TVf{q) - TVq ./ (q)}. 
In an exactly similar manner we should obtain 

df{q) = dq .f (q) +dUVq . [TVf(q) - TVq ./' (q)}. 

If UVq be constant, i.e. if the vector of q preserve a constant 
direction, then 

dUVq = and VUVq=0, 

and in that case we have the two relations 

Y/(<Z) = Vq .f (q), df(q) = dq .f (q). 

The equation dUVq = requires that UVq = const., and it is 
easily established that if q is to be a real quaternion then the 
equation V UVq = also requires the same condition. 

4. Instead of the elementary solutions of V?* = used in my 
former paper, we might have taken the simpler pair 

u = y + kx, v = z —joe, 

in which case our theorem would have taken the form 

7 , dr , dr 

dr — du .;r-+dv.~-. 
oy dz 



Now, if \ and fi be scalar constants 
UV(Xu+fxv) = 



\k — /jij 



and is, therefore, constant. We also have 

V (\u + fiv) = 0, 

and it, therefore, follows by the preceding article that 

V . e Au +' xy = 0. 

Thus we see that the general integral of the equation Vr = 
may be written in the symbolical form 

ul + vS- 

r = e f{\, p), 



154 Mr Brill, On Quaternion Functions with especial [Feb. 23, 

where 

f(\ H ,) = A+B\+Cfj, + ^ 1 {D\ 2 + 2EX/J, + F^} 

+ i {GX 3 + SH\y + SKXfj? + Lfi 3 } + &c., 

the coefficients A, B, C, &c, being in general quaternions. The 
generality of this form will not be affected if after expansion of the 
exponential symbol and subsequent differentiation we make X and 
/a both zero. Hence we obtain 

r = A + uB + vC + ^ {u 2 D + (uv + vu) E + v 2 F] 

+ ^ [u*G + {a 2 v + uvu + vu 2 ) H + (uv 2 + vivo + v 2 u) K + v*L) + &c. 

It is important to notice that the quaternion coefficients must 
always be placed last in the various terms of the series. Graves 1 
gave a form similar to this as a quaternion solution of Laplace's 
Equation, but he seems to have only contemplated the possibility 
of scalar coefficients, and in consequence his result loses in 
generality. The importance of introducing quaternion coefficients 
will be at once seen if we attempt to express the elementary 
solutions mentioned in my former paper in terms of those used in 
this. We have 

— 2ix + jy + kz = (y + koo)j + (z —joe) k, 
ix — 2jy + kz = - 2 (y + kso)j + (z —joe) k, 
ix+ jy — 2kz = (y +kx)j — 2 (z — jx) k, 
2(x + y + z) + i(y — z) +j (z — x) + k (x - y) 

= (y + kx) (2+i-k) + (z -jx) (2 - i +j). 
By means of the equation 

df(q)=dq.f'(q) 
of the preceding article, which holds in the case under con- 
sideration, we have 

dr = (du -^+ d v.f)e ^ ^f(X, fi) 2 

7 u^ + v^df 7 ^i + v^df 

= du.e dA w^r+dv.e <* ^4~ 
oX dfi 

= du . U+dv . V, 

1 "On the Solution of the Equation of Laplace's Functions," Proc. B. I. A., 
vi. 162—171, 186—194. Graves's papers were written with the object of giving 
the interpretation of a symbolical form obtained by Carmichael, " Laplace's Equation 
and its Analogues," Cambridge and Dublin Mathematical Journal, vu. 126 — 137. 

2 It is to be understood that after expansion of the exponential symbols and 
subsequent differentiation, \ and /j. are to be made zero. 



1891.] Reference to the Discussion of Laplace's Equation. 155 

where U and V are formed from of/d\ and df/djj, in a similar 
manner to that in which r is formed from f(\,, //.). Thus we see 
that if we take u and v as our fundamental solutions of Vr = 0, the 
formula discussed in my former paper is very closely analogous to 
the formula 

7 dw lf . dw , 
dw = yp d% + ^— d-q, 

which would hold if w were a scalar function of the two variables 
£ and y. 

Instead of the pair of special solutions that we have here made 
use of, we might have chosen either of the pairs 

z + ixj and x — ky, 

x +jz and y — iz. 

I have investigated the matter and find that, adhering to real 
quaternions, the most general form that we can take for our pair 
of special solutions, in order that a third solution may be expressed 
in terms of them in the simple form of the present article, can by 
a suitable choice of axes be expressed by 

y + lex, y cos a + z sin a — x (j sin a — k cos a). 

5. The expression for r given in the preceding article is 
obviously not perfectly general, as it is derived from a series con- 
taining only positive integral powers. In the second part of the 
paper referred to above, Graves gave a method of deriving solutions 
from scalar series containing negative and fractional powers, but 
he did not succeed in expressing his results in terms of the two 
special solutions he made use of. I think that it is highly probable 
that if we take any two independent special solutions, any other 
solution can be derived from them ; but at the same time it is 
possible that the said other solution may not be expressible in 
terms of the two special solutions in the ordinary functional 
form. 

[There is one remark to be made in completion of my former 
paper. It is there proved that if p and q be any two inde- 
pendent solutions of the equation Vr=0, and r any other solu- 
tion, then there exists a relation of the form 

dr — dp.P + dq.Q, 

where P and Q do not involve the ratios dx : dy : dz. The converse 
of this is also true : for since the above relation is to be satisfied 



156 Mr Adami, On the disturbances of temperature [Mar. 9, 

for all small variations of the point (x, y, z), it involves the three 
relations 

dx dx' dx ' ' 

?!:=§£ p + h n 

dy dy' dy ' 

dz dz' dz' 

Hence it follows that 

Vr = Vp . P + Vq . Q, 
so that if Vp = and V^ = 0, we have Vr = also.] 

[It is to be remarked that a theory similar to the one 
established in the above paper can be constructed for the equation 
discussed in Art. 5 of my former paper, by choosing for the 
elementary solutions the expressions 

x — it, y —jt, z — kt. 

April 27th, 1891.] 



March 9, 1891. 

Dr Lea in the Chair. 

H. G. Dawson, M.A., Christ's College, was elected a Fellow of 
the Society. 

The following Communications were made to the Society: 

(1) On the disturbances of the body temperature of the fowl 
which follow total extirpation of the fore-brain. By J. George 
Adami, M.A., M.B., Christ's College, John Lucas Walker Student 
in Pathology. 

[Abstract; received May 8, 1891.] 

In the course of a series of experiments, undertaken at Paris, 
upon the development of fever by means of aseptic solutions 
of the products of bacterial growth it became important to take 
into consideration how far the various phases of the febrile state 
depend upon the action of the higher nervous centres; to see 
whether a typical fever can be induced when the cerebral hemi- 
spheres have been removed, and to investigate the terms of 
relationship between ' the heat-centres ' (if these have a local 
existence) and the altered conditions of heat production and 
the giving off of heat which obtain during the febrile state. 



1891.] which follow total extirpation of the fore-brain. 157 

These questions I do not propose to answer in the present paper, 
which is but of the nature of a preliminary communication. 
I can here only state the results of certain first steps towards a 
resolution of the problems, and note the variations in the tempe- 
rature of the body, as measured by the thermometer, during the 
days immediately following upon the removal of the cerebrum. 

For these experiments I made use of the fowl. Mammals 
were out of the question for any prolonged observations, inasmuch 
as they cease to manifest any vitality whatsoever in the course 
of but a few hours after the hemispheres have been extirpated ; 
and I employed the fowl rather than the pigeon (which other- 
wise has many advantages) because with the former it is the 
more easy under ordinary conditions to induce experimental fever. 
In the a-volitional non-sentient state which follows removal 
of the cerebral hemispheres the fowl, it is well known, may con- 
tinue for weeks, and it may be months. When however in 
addition to the hemispheres the optic lobes are to a greater or 
less extent extirpated, as was purposely the case in my ex- 
periments, then this state, it would seem, can last for a much 
shorter time, the bodily functions ceasing in from one to ten 
days according to the amount of obliteration of these organs 
that has been practised. Hence in these experiments it cannot 
be said that the stage of ' shock ' following the operation has 
definitely been passed : it is impossible to declare that the 
variations in temperature which I am about to describe are not 
largely due to the highly irritable condition of the rest of the 
nervous system brought about by operative interference and re- 
moval of the higher centres. 

Kept at an equable and moderate external temperature the 
ordinary fowl exhibits during the day a variation in the body 
temperature of at most 0*75° C, the mean temperature of well- 
fed fowls as measured in the rectum being — in winter — about 
42 - 3° C. (108° F.). But after extirpation of the hemispheres and 
optic lobes — the latter wholly or partially — the temperature 
variations became very wide, passing from below 35° to above 
45° C, and it was a matter of extreme difficulty to prevent, 
even for a few hours, well-marked ascents or descents of the 
temperature. Removal, therefore, of so large a portion of the 
brain had thoroughly disturbed the balance between the thermo- 
genic and thermolytic powers of the organism. 

So great had been the disturbance that now the fowls reacted 
to changes in the external temperature much in the same way 
as do cold-blooded animals. Placed in a room whose temperature 
was 22° C. (71 "6° F.), and covered carefully with cotton wool 
the rectal temperature rose rapidly until in those instances in 
which the rise was unchecked it reached the height of 44'5° C. 



158 Mr Adami, On the disturbances of temperature [Mar. 9, 

(1120° F.) or more, in one case becoming as much as 45'325° C 
(113'58° F.). Removal of the cotton wool checked the rise and 
often induced an actual fall through more than a degree in the 
course of two hours. With an atmosphere slightly warmer 
(24° C) no cotton wool was necessary to bring about an ascent. 
Transference to a room whose temperature was some degrees 
below 22° C led to a rapid lowering of the point to which the 
mercury rose. Thus in one case, the fowl being placed in a room 
at 24° C. and covered with wool, the body temperature rose four 
degrees in five hours, from 40*2° to 44'2° C. ; transferred to a room 
at 18° the animal shewed in less than three hours a fall through 
5*6 degrees, to 38'6° ; then replaced in the room at 24°, this time 
without a covering of wool, the temperature rose slowly but 
steadily until it reached 42 - 8° at the end of eight hours. I might 
adduce many other instances to the same effect. 

Similarly, in animals whose fore-brain had been extirpated, 
15 ccm. of cold water poured into the crop caused a fall of from 
half a degree to a degree during the course of the succeeding 
half-hour. This amount of cold water has no effect upon the 
body temperature of the normal fowl. On the other hand a rich 
proteid diet in the form of an egg, beaten up and warmed to 
nearly the temperature of the body, caused invariably a well- 
marked rise of one to three degrees beginning two to three 
hours after the animal had been fed. This rise reached its 
maximum in about six hours, and may be compared to the rise 
that has been found to occur in the crocodile and, I believe, 
the snake, after a large meal of animal food. 

It may be noted here that while change to a cooler atmosphere 
caused lowering of the body temperature, this lowering, if the 
change had not been too considerable, tended to give place 
eventually to a slow rise. To this extent the reaction differs 
from what obtains in the cold-blooded animal, and despite the 
removal of the fore-brain there would seem to be a tendency for 
the body temperature to be brought back to the normal. 

With the temperature liable to such great and constant 
fluctuations it was extremely difficult to determine the effects 
of injections of fever-producing substances, as, for example, 
sterilised bouillon in which the Vibrio Metschnikovi had been 
grown, or to know at what moment these might be made. Never- 
theless in the two cases in which such injections were performed 
under what appeared to be favourable conditions there was so im- 
mediate and steady a rise of temperature through two degrees 
during the succeeding eight hours, that I am led to see in this 
rise an indication that febrile changes may be induced in the 
hen deprived of its hemispheres, and, if it be accepted that in 
the fore-brain lies the main heat regulating mechanism of the 



1891.] which follow total extirpation of the fore-brain. 159 

body, that febrile changes may be induced independently of this 
mechanism. 



(2) On the nature of Supernumerary Appendages in Insects. 
By W. Bateson, M.A., St John's College. 

[Abstract.] 

The author exhibited a number of specimens in illustration of 
this subject. 

The evidence related to about 220 recorded cases of extra 
legs, antennas, palpi or wings, and particulars were given as to 
the mode of occurrence of these structures. 

Speaking of cases in which the nature of the extra parts could 
be correctly determined, it was found that the following principles 
were generally followed : 

i. Extra appendages arising from a normal appendage usually 
contain all parts found in the normal appendage peripherally to 
the point from which they arise, and never contain parts central 
to this point. 

ii. Such appendages are commonly double. The axes of the 
three appendages then stand in one plane, one being nearer to the 
normal appendage and one remote from it. In structure and 
position the nearer limb is the image of the normal limb in a 
mirror perpendicular to the plane in which the limbs stand, while 
the remoter extra appendage is the image of the nearer one in a 
remote mirror parallel to the first. Thus if the normal limb is a 
right limb, the nearer supernumerary is a left and the remoter a 
right, and vice versa. 

An extra appendage sometimes occurs which is apparently a 
single structure. In all instances in which the matter could be 
determined, it was found that the apparently single appendage in 
reality consisted either of two anterior halves or of two posterior 
halves of a pair of appendages conforming to the law stated. 
Probably therefore no extra appendage is morphologically single. 

It was pointed out that these phenomena are important as 
an indication of the physical nature of bodily symmetry, and 
in their bearing upon current views of the character of germinal 
processes. 

The author expressed his indebtedness for information, or the 
loan of specimens, to Messrs H. Gadeau de Kerville, Pennetier, 
Giard, Kraatz, L. von Heyden, Dale, Mason, Westwood, Water- 
house, N. M. Richardson, Janson, Reitter, &c, and especially 
to Dr Sharp for much help and advice in examining the 
specimens. 



160 Mr Groom, On the Orientation of Sacculina. [Mar. 9, 

(3) On the Orientation of Sacculina. By Theo. T. Groom, B.A., 
St John's College. 

[Abstract; received March 10, 1891.] 

In comparing the larval stages of various members of the 
group Cirripedia I have found it necessary to come to some inde- 
pendent conclusion as to the relation of the adult animals in the 
several groups to one another. One of the questions raised con- 
cerns the morphology of the Rhizocephala. 

The Rhizocephala are, as is well known, a group of small 
animals parasitic on Decapod Crustacea. 

It is only of late years that we have obtained anything like 
an accurate knowledge of the structure and life-history of any 
member of this group; but since the appearance of Delage's 
classical paper 1 on the development of Sacculina we have had a 
fair knowledge of one species. 

Two views as to the orientation of Sacculina have been main- 
tained, the earlier by Kossmann' 2 , and the later by Delage. 

In order to determine the orientation of Sacculina it will be 
necessary to briefly compare the structure of the adult with that 
of a typical Thoracic Cirripede such as Lepas or Pollicipes. 

In both, the whole structure indicates a primitive bilateral 
symmetry on each side of a median plane. At one end of the 
body is a peduncle or more or less elongated stalk by which the 
animal is attached : this in Sacculina, in accordance with its para- 
sitic mode of life, gives off rootlets for the absorption of nutriment. 
The body gives off in the median plane in close relation with the 
peduncle at one point the mantle, the histology of which presents 
considerable similarity of a special character in the two forms, 
as seen in the accounts of Delage in Sacculina and of Koehler 3 
and Nussbaum 4 in Lepas, Pollicipes, etc. The differences in histo- 
logy between the two are evidently closely connected with the 
absence in Sacculina of the strong calcareous plates of Lepas and 
the consequent predominance of muscular and connective tissue 
elements, a result of the different modes of life of the two forms. 
The mantle surrounds the body on all sides, forming the mantle- 
cavity closed except at the end remote from the peduncle, where 
the mantle-opening leads to the exterior. The mantle-cavity has 
in both the same function of retaining and probably of aerating 
the eggs. The mantle is attached to the body by a band of tissue 
not distinct from the peduncle in the adult Lepas (although 

1 Evolution de la Sacculine. Archives de Zool. Exp. et. Gin., 2 Serie, Tome 2, 
1884. 

2 Suctoria und Lepadidae. Arbeiten a. d. zoolog. zoot. Institut in Wiirzburg, 1, 
1874. 

3 Recherches sur l'organisation des Cirrhipedes. Archives de Biologie, ix, 1889. 

4 Anatomische Studien an Californiscben Cirripedien, Bonn, 1890. 



1891.] Mr Groom, On the Orientation of Sacculina. 161 

definitely recognisable in the larva) but forming a membrane in 
Sacculina running from the peduncle to the mantle-opening and 
termed by Delage the mesentery. Symmetrically situated on each 
side of the median plane are the oviducts opening into the mantle- 
cavity by the two female genital pores*. The relation of the 
oviduct to the ovaries is similar in both cases, though in conse- 
quence of the different situation of the ovaries (due perhaps to 
the special mode of nutrition in Sacculina) the relative lengths 
are very different : in both, the lower part of the oviduct is 
expanded to form a chamber 2 , the glandular walls of which, 
simple in Lepas, but branched in Sacculina, constitute the 
Kittdrilsen of the Germans and "Glandes cementaires" of Delage 
(as Giard 3 has pointed, out and Delage admits, these glands have 
no relation with the cement-glands proper of Cirripedia). The 
function of these glands in both groups is to produce the peculiar 
sac or cocoon 4 in which the ova are enclosed when lying in the 
mantle-cavity as the ovigerous lamellae of Darwin 5 . 

The openings of the vasa deferentia (also symmetrical) in 
Cirripedes are less constant in position than the other organs 
mentioned, and I wish to reserve all mention of them for a future 
occasion. 

The last structure to be compared in the two forms is the 
nervous system. This in Lepads consists of a supra-oesophageal, 
a large sub-cesophageal and a series of posterior ganglia. In Sac- 
culina a single ganglion is present, corresponding probably in 
position with the sub-cesophageal ganglia of Lepads or Balanids, 
both being near the female genital apertures. 

Now both Kossmann and Delage are agreed that the plane of 
symmetry passing through the mesentery in Sacculina is the 
median vertical plane. Kossmann places the peduncle in front, 
the mantle-opening behind and the mesentery on the dorsal line. 
Delage making the mesentery ventral and the mantle-opening 
posterior gives Saccidina a position diametrically opposite to that 
of Kossmann. He bases his view solely on the situation and 
origin of the ganglion. Since, he argues, this arises on the ventral 
side in articulate animals, the neighbourhood of the edge of the 
mesentery close to which the ganglion originates must also in 
Sacculina be ventral ; in other words the mesentery is ventral ; 
and since the nervous centre in Cirripedes is situated in the 
region of the thorax, the ganglion of Sacculina must indicate the 

1 Krohn, Wiegmann's ArcMv. f. Naturc/eschichte, 1859. Hoek, Challenger 
Report, Vol. x., Cirripedia, Anatomical Part, 1884. Nussbaum, loc. cit. 

2 Hoek, Nussbaum, loc. cit. 

3 Sur l'orientation de Sacculina Carcini. Comptes Rendus, 102. 

4 Hoek, Nussbaum, loc. cit. 

5 A monograph of the Cirripedia, Lepadidae and Balanidae, Eay Society, 1851, 
1854. 



162 Mr Groom, On the Orientation of Sacculina. [Mar. 9, 

point where the thorax would be if present ; i. e. the posterior end 
of the animal. 

I must admit, however, that it seems to me quite impossible 
to determine the orientation from one point alone, and that 
given the ganglion were situated at the posterior end of the 
body on the ventral side, I fail to see why the side on which the 
mesentery is situated cannot be equally dorsal or ventral. We 
need more than one point to determine the orientation of any 
animal, and this it seems to me is given in the present case by 
the comparison of other structures. Delage, however, rejects the 
evidence furnished by the other organs, and bases this rejection 
on the embryology of Sacculina, of which he has given so inte- 
resting an account. He finds that upon the fixation of the 
Cypris-form the young Sacculina loses all its organs (carapace, 
appendages, etc.) and becomes reduced to a mass of embryonic 
cells from which all the organs of the adult Sacculina develop de 
novo. He concludes from this that not only are the organs of the 
adult morphologically different from those of the Cypris-stage, 
but that there is no necessary agreement of any one of the sur- 
faces of the adult with any one of the larva, the former being 
determined wholly by the relation of the parasite to the host. 

The development is certainly remarkable, but I think there is 
no reason to doubt as Giard 1 has done the general correctness of 
Delage's account. The development of organs de novo occurs, 
however, in other forms and the case in point seems to me hardly 
more marvellous than the re-development of the three posterior 
pairs of maxillipedes in Squilla after they have been once lost 2 , 
or the similar reproduction of the last two pairs of thoracic 
appendages in Sergestes 3 , and other analogous cases : yet few 
would venture to doubt the homology of these appendages with 
the corresponding ones in allied forms 4 . 

The tendency of late years has been, I think, to admit that a 
very considerable amount of modification of the ancestral develop- 
ment may take place, and that we must be exceedingly cautious 
before admitting any case of ontogeny as presenting a truthful 
representation of the phylogeny. The comparison of the structure 
of the adults will in many cases be of greater service. 

It appears to me, therefore, that the cumulative evidence of 

1 Giard, loc. cit. 

2 Balfour, A Treatise on Comparative Embryology, vol. i., 1880. 

3 Balfour, op. cit. 

4 I must point out that the nervous centre is one of those organs which, accord- 
ing to Delage, arises de novo, and that if such organs are not to be regarded as 
homologous with those of the larva, Delage is hardly justified in determining the 
orientation of Sacculina by means of the position of the ganglion, since it can 
hardly be doubted that the nervous system of the Sacculina Cypris-form is the 
same thing as that of the Cypris-form and adult of Thoracic Cirripedes. 



1891.] Mr Groom, On the Orientation of Sacculina. 163 

the organs compared in Lepas and Sacculina speaks very strongly 
in favour of their homology in the two, and if this be the case 
there can be little doubt as to the true orientation of Sacculina, 
and that though the supposed development on which Kossmann 
partly based his case has been shewn by Delage not to take place 
it will be necessary to return to the older view based upon less 
complete information, that the mesentery is dorsal and the 
mantle-opening posterior. 

There is one more point to which I wish to draw attention. 
The great similarity in structure and relation of the oviduct in 
Lepads and in Sacculina makes it exceedingly probable that these 
organs are truly homologous. Now in all Cirripedes in which 
the position of the genital pores has been accurately determined 
they are very constantly situated at the base of the first of the 
six pairs of cirri 1 . In all probability then the portion of the 
body on which the two oviducts open is thoracic. This conclusion 
agrees well with the position of the ganglion which corresponds 
closely with that of the largest ganglion of the Thoracica. I 
would suggest then that when fixation of the Cypris-form takes 
place a great reduction in size of the thorax occurs at the same 
time as the moult accompanying that fixation, and that the whole 
of the thorax and abdomen is not cast off in the way Delage 
supposes, but that a portion, small perhaps, but morphologically 
representing them is left attached to the head. This would indicate 
a conception of the Rhizocephala differing considerably from that 
proposed by Delage who considers the adult Sacculina to represent 
the head alone of other Crustacea. The visceral portion of the 
parasite would consist of head and thorax fused together; the 
latter, judging from the position of the female genital pores, 
occupying as much as one-quarter of the whole. The carapace 
attached to the dorsal side of the head in Lepas and in the Cypris 
stage of Sacculina must then be regarded as fused to the dorsal 
side of the thorax as well as the head, in a manner analogous to 
that of the higher Crustacea. 

(4) Some experiments on blood-clotting. By Albert S. Grun- 
BAUM, Gonville and Caius College. 

[Abstract; received March 3, 1891.] 

In the ' Centralblatjb fur Physiologie' of 1888, p. 263, there is 
a paper by K. Bohr on the respiratory changes due to injection of 
peptone and leech-extract, and in this paper he incidentally 

1 I have lately been able to supply the only missing link in the chain of 
evidence which points to Darwin's "Acoustic sacs" as marking the termination of 
the oviduct : the eggs in one specimen of Lepas were found actually issuing from 
the orifice of the sac. 

VOL. VII. PT. IV. 14 



164 Mr Larmor, On the most general type of [May 4, 

mentions that if, in the rabbit, the coeliac and superior mesenteric 
arteries be tied, the blood, if collected after about four hours, will 
not clot for about two hours. He intimates his intention of en- 
quiring further into this matter, since his assertion is based on 
two experiments only; but he has as yet written nothing more 
on the subject. 

In April, 1890, I commenced repeating his experiments, but 
I have not yet had time to investigate the matter completely. 
However, in six experiments in which the coeliac and superior 
mesenteric arteries were ligatured, the blood collected always 
clotted in less than two hours : in fact, generally under ten 
minutes. In one case only were twenty minutes required before 
the clotting became evident. It is true that the time was slightly 
lengthened, since, normally, clotting takes place in 3-5 minutes ; 
and the clot was decidedly not so firm as usual. The experiments 
were performed as Bohr describes, except that in the first two, 
the abdomen was opened in the middle line, instead of at the 
side. After four hours, the animal being under chloroform during 
the whole time, the blood was collected from the carotid. Bohr 
states that his animals were apparently as well at that time as at 
the beginning : I always found the stomach partially self-digested 
and the intestines becoming gangrenous, and there was a great 
fall of temperature. 

The occlusion of the arteries by the ligature was always tested 
by injecting the animal after death, and, with one exception in 
which the coeliac artery let the injection through, was found to be 
complete. Thus it appears that some slight though evident effect 
on the clotting may be produced by stopping the circulation 
through the stomach and intestines, but it is generally not as 
marked as in Bohr's two experiments. 



May 4, 1891. 

Peof. G. H. Darwin, President, in the Chair. 

The following Communications were made to the Society : 

(1) The most general type of electrical waves in dielectric 
media that is consistent with ascertained laws. By J. Larmor, M.A., 
St John's College. 

It was explained that Maxwell's hypothesis of complete circuits 
reduces the whole of electrodynamics to the ascertained Ampere- 
Faraday-lSfeumann laws for such circuits, as it completely defines 
the character of the electrostatic polarisation which must be 
postulated as a part of the theory. The question as to how far 
the theory of electrodynamics may depart from this simple form 



1891.] electrical waves in dielectric media. 165 

when the circuits are not assumed to be complete was then 
examined mathematically on the lines of v. Helmholtz's investiga- 
tions. It was shown that waves of transverse displacement will 
always be propagated in a dielectric, irrespective of what hypo- 
thesis is assumed as to the law of the mutual action of incomplete 
currents, whether that adopted by v. Helmholtz or the still 
more general one which is formally possible. But it was also 
shown that, if the velocity of these waves in a non-magnetic 
dielectric is equal to the inverse square root of the specific 
inductive capacity, the currents are necessarily complete. The ex- 
perimental evidence is strongly in favour of this relation, and in 
so far constitutes a demonstration of Maxwell's theory of electro- 
dynamic action and its mode of propagation in stationary media 1 . 

(2) A mechanical representation of a vibrating electrical system, 
and its radiation. By J. Larmor, M.A., St John's College. 

The propagation of undulations of electric polarisation in a 
dielectric is exactly similar to the propagation of elastic waves 
in a solid medium, which must be absolutely incompressible if 
we follow Maxwell's scheme, but may transmit waves of con- 
densation as well as shearing waves if we admit the more general 
scheme developed by von Helmholtz. 

The propagation of electrical actions in a medium which is a 
conductor of ordinary type (so that the number which expresses its 
inductive capacity in C.G.S. electromagnetic measure is very small, 
while that which expresses its specific resistance is large) follows 
the same law as the diffusion of heat in a conducting medium, 
with the proviso that the thermal diffusivity is to be proportional 
to the reciprocal of the electric conductivity. For this reason 
rapidly alternating disturbances in the medium surrounding a 
conducting mass are only transmitted skin deep into the con- 
ductor, the depth of this skin diminishing with increasing rapidity 
of the alternations and increasing conductivity, in the same manner 
as in the corresponding question of the propagation beneath 
the surface of the ground of the daily and annual alternations of 
temperature. For this reason also a sheet of conducting metal acts 
as a screen against alternating electrodynamic influence. 

In development of this elastic solid analogy it has recently 
been explained by Sir W. Thomson 2 that the magnetic induction 
due to a steady electric current traversing a circuital channel 
in the medium is represented by twice the vorticity of the elastic 
strain when a longitudinal force of amount represented by the 
current is applied to the portion of the medium which coincides 
with this channel. 



1 See Proc. Roy. Soc. 1891, Vol. xlix., p. 521. 

2 Math, and Phys. Papers, Vol. in. p. 451. 



14—2 



166 Mr Larmor, On a mechanical representation of a [May 4, 

It is also a well-known relation, and forms in many respects 
the simplest and most symmetrical mathematical specification 
of the connexions of the electric field on Maxwell's scheme, that 
the magnetic induction (abc) is represented by the vorticity of 
the electric force (PQR), and the electric force by the vorticity 
of the magnetic induction, according to the equations 

dQ dR _ da 
dz dy dt ' 
db dc . ( 7r d \ D 

n -T y =- *"* r j t + v p - ■••■•••■ 

In the dielectric, a is zero, and the electric displacement (fgh) is 
proportional to the electric force according to the relation 

K d 
Thus the vorticity of the electric displacement is — -j- of the 

magnetic induction ; and the time integrals of (fgh) represent 
on Sir W. Thomson's analogy the displacements of the elastic 

medium multiplied by the factor ^— • 

Or we may, as in the following sections, take the electric dis- 
placement to represent the actual displacement of the elastic 
medium, and then the magnetic induction will be equal to the 

8-7T 

time integral of its vorticity multiplied by ~r ; and the electric 
current will be equal to the time integral of the impressed 

4*7T 

forcive multiplied by -^- . This forcive must on Maxwell's 

scheme be circuital, that is, circulating in ideal channels in the 
manner of the velocity system of an incompressible fluid. 

These considerations thus present a mechanical view of the 
electric propagation in dielectrics, with the exception that the 
current on the wire must be imitated by an applied forcive of 
some kind. If the media are magnetic, differences in rigidity 
or of density must be introduced between them. 

For an electrical vibrator we can however complete the me- 
chanical analogy, provided the wave-length of the undulations 
is not smaller than a few centimetres in the case when the 
vibrator is made of an ordinary conducting metal. For ex- 
perimental types of Hertzian vibrators the analogy will therefore 
be practically exact. 

To do this we have to examine the conditions as regards electric 
displacement that must be satisfied at the surface of a conductor. 



V, 



1891.] vibrating electrical system, and its radiation. 167 

The displacement is proportional to the electric force, which 
consists of a kinetic and a static part. The kinetic part is 

-!«<?, a). 

where V (F, G, H) = — ^tt/j, (u, v, w) 

because FGH is the vector potential of the current distribution ; 
it is therefore continuous everywhere as the potentials of all 
volume or surface distributions must be. The static part 

d d d 
^dx ' dy ' dzj 

can only be due to a surface density s on the conducting surface; 
therefore its components along the surface must be continuous 
when we cross it, while the discontinuity in the component normal 
to the surface must be equal to 4<7rs. The tangential components 
of the total electric force are therefore continuous. 

Now it is known that for rapid alternations the currents are 
induced only in the skin of the conductor ; the vibrations of the 
system and its free periods are in the limit quite independent of 
the conductivity and of the nature of the interior parts of the 
conductor : they are practically the same as if the conductivity 
were infinite, and there is no sensible decay owing to any degrada- 
tion into heat. 

We require the proper boundary conditions to express these 
facts. 

The currents in the conductors may be treated as surface 
sheets inside which the electric force is zero. 

Outside the sheet therefore the components of the electric 
force along the surface must be zero also ; and therefore the 
tangential electric displacements must be zero. 

The components of the electric force normal to the surface 
will differ on the two sides of it by an amount determined by the 
surface density s. 

In the elastic solid analogy we may therefore consider the 
conductors to be cavities in the solid, provided we confer infinite 
rigidity on the skin of each cavity so that no point of it can have 
any tangential displacement. If we assume that the solid is 
incompressible, its vibrations are completely determined by these 
conditions ; the component of the displacement normal to the 
surface must naturally adjust itself in such manner that the 
condensation remains always null. This will be the analogue of 
an electric surface density on the conductor which adjusts itself 
instantaneously to the equilibrium value corresponding to the 
actual phase of the disturbance in the dielectric, according to 



168 Mr Larmor, On a mechanical representation of a [May 4, 

the equation V 2 V = which of course indicates infinite velocity of 
propagation or adjustment of disturbances. 

The medium being incompressible, the total volume of all 
the cavities will remain the same. We may put this necessary 
property of the motion in evidence by supposing each cavity to 
be filled with incompressible fluid. The displacement of this 
fluid at the surface will then be continuous with the displace- 
ment of the solid and will represent the electric surface density. 
The inertia of the fluid must not sensibly interfere with the 
adjustment of the normal displacement ; so that if the rigidity 
of the solid is taken to be finite, the fluid must be of negligible 
density. But it is to be borne in mind that, as the fluid re- 
presents a conductor, the motion of the fluid does not represent 
electrostatic displacement except on the surface. 

We may thus represent the circumstances of an electric vi- 
brator by the annexed diagram. 




Two condensers A, B, are represented, with their inner coatings 
connected by a conducting wire in which the spark gap required 
for the production of the initial disturbance is usually situated ; 
and their outer coatings are connected by another wire which 
may be to earth. Each conducting system forms a cavity in the 
elastic dielectric, which may be considered to be filled with in- 
compressible massless liquid ; in this case there are two such 
cavities with pairs of plane faces opposed to each other. 

The disturbance may be supposed to be originated by getting 
an excess of liquid into the inner coating of the condenser A ; 
that involves pushing away the plate of dielectric between the 
two coatings of this condenser, and therefore removing an equal 
amount of liquid from the outer coating. For the parts of the 
conducting surfaces other than the opposed faces are all backed 
up by thick masses of dielectric, which will not yield sensibly as 
compared with the thin plates of dielectric which belong to the 
condensers. The greater mobility of the latter accounts both 
for the store of energy that the condenser can acquire, and for 
the equality of the charges of its two coatings. 



1891.] vibrating electrical system, and its radiation. 169 

When the system is disturbed the liquid will sway backwards 
and forwards between the condenser coatings in a way which 
gives a very real representation of the actual electric oscillation. 

The only element in this representation that is not easily 
realisable is the skin, of rigidity great compared with that of 
the solid ; if however the solid were a jelly the skin would be 
naturally provided by supposing the system represented in the 
diagram to be constructed of very flexible sheet metal. 

The greater the capacity of the condensers compared with the 
section of the connecting wire the longer is the period of the 
graver vibrations ; thus illustrating the dependence of the period 
on the capacity and self-induction of the vibrator. There are 
also overtones in which the pulse of normal displacement is 
almost confined to the connecting wire, the greater mobility at 
its ends making those points approximately nodes ; and their 
nodal character will be the more prominent the greater the 
capacity per unit area at the points where they are attached to 
the condensers. For these overtones the half wave-length is thus 
a sub-multiple of the length of the connecting wire. 

Owing to the small surface of the connecting wire these over- 
tones would not have much chance of being communicated to the 
surrounding medium, except by reason of the general principle 
which requires that the shorter the period for given dimensions of 
the vibrator the more of the vibrational energy travels outwards 
into the medium, and that in a ratio which increases very rapidly 
with increasing frequency. In the Hertzian oscillator the con- 
densers are replaced by large metallic plates, and it seems clear 
that the waves that are the chief subject of experiment issue from 
the large surfaces afforded by these plates, and so belong to the 
lower periods of the vibrator. 

The waves of a Hertzian vibrator are therefore radiated from 
the plates of the vibrator : to obtain considerable radiation to a dis- 
tance it is essential that the dimensions of a plate should be con- 
siderable compared with the length of a wave of the radiation. 
This condition would not be fulfilled if the plates were replaced by 
condensers, for though the energy of the vibration would thereby 
be increased, its wave-length for the fundamental periods would be 
lengthened, and the radiation, therefore, very much diminished. 
Condensers would be allowable instead of plates when the disturb- 
ance is guided along conducting wires, but as a rule they would 
give only slight undulations in the dielectric at a distance from 
conductors. 

The plates are thus like two poles radiating in opposite phases : 
and the general equations given by Kirchhoff to represent this 
type of motion are those used by Hertz in his discussion of the 
general character of the radiation at a distance from the vibrator. 



170 Mr Larmor, On a mechanical representation of a [May 4, 

[The different types of vibration that may theoretically exist 
would therefore appear to be as follows. A reciprocating flow 
may be set up between the plates, along the connecting wires ; if 
the capacities are at all considerable its period will be compara- 
tively slow, being calculable from capacity of condenser and self- 
induction of wire alone, because there will be involved very little 
disturbance in the dielectric except the static value of the dis- 
placement at each instant, and there will be no sensible amount of 
radiation. An oscillation of the dielectric to and fro between the 
two plates may be set up, corresponding to a very small period, 
the wave-length being twice the distance between the plates in the 
case in the diagram, when the earth connexion is good ; this will not 
be sensibly affected by the presence or nature of the connecting 
wire, and it will involve rapid decay by radiation, but it will 
probably be very difficult to excite to any sensible amount. And 
there may be superficial dielectric waves running along the con- 
necting wire, of period about the same as the preceding when the 
wire is straight. 

For the linear type of vibrator consisting of two equal cylinders 
with a spark gap between them, and no capacities at the ends, 
the wave-length would be the length of the vibrator simply, as 
from the mode of excitation its two halves would always be in 
opposite phases.] 

The nature of the approximations involved in this method of 
representation will be sufficiently put in evidence by the following 
investigation of the circumstances of the reflexion of a system of 
waves from a metallic plate. It will be found that for wave- 
lengths of a centimetre or more the reflexion of all ordinary 
metallic plates is sensibly perfect, and involves an acceleration of 
phase of half a wave-length. For smaller wave-lengths, corre- 
sponding to those of light waves, the circumstances of the reflexion 
are more complicated. 

We shall proceed on Maxwell's theory, which postulates the non- 
existence of condensational effects : no other theory can make the 
velocity of propagation of waves of transverse displacement in- 
versely proportional to the square root of the specific inductive 
capacity of the medium 1 . 

The first point is to assume precise definitions of the quantities 
that enter into the equations. With the usual notation 

dG dF , 

c = fa-dt> a = -" b = -> 

so that, if J denote 

dF dG dH 

dec dy dz ' 

1 See Proc. Roy. Soc. 1891, Vol. xlix. p. 521. 



1891.] vibrating electrical system, and its radiation. 171 

we have 

d J _ db da 
dx dx dy 

= — 4f7T/MU. 

Hence defining FGH by the formula 

F, G, H = j—fi(u, v, w), 

we annul J, since 

du dv dw _ 
dx dy dz 

owing to the absence of condensation. 

The components of electric force are 

dF dV 
r ~~ dt dx' ¥ — ■»-« — ■» 

where V is a function whose presence is necessitated by the fact 
that the currents must be circuital ; it must enter in this form in 
order that it may produce no result on integration round a circuit, 
and it is completely determined by the conditions of any special 
problem. The fact that V must be a single-valued function so as 
to give a null result on integrating round a circuit shows that it may 
be expressed as the potential of an electrostatic distribution ; thus 
its introduction into the equations is accounted for consistently 
with ordinary electrical ideas. 

In a dielectric the total current uviv is the displacement current 
in the dielectric, so that 

K d 

(u, v ^ w ) = ^ Tr j t ( p > Q> R )- 

In a conductor the displacement current is evanescent com- 
pared with the current conducted according to Ohm's law, so that 

a (u, v, w) = (P, Q, R), 

where a is the specific resistance of the medium. 

Also, by definition above, 

V 2 (F, G,H) = - 47r/i (u, v, w). 

Thus in a dielectric the equations of propagation of the vector 
FGH are of the type 

fiK df dxdt ' 



172 Mr Larmor, On a mechanical representation of a [May 4, 

in a conductor, they are of the type 

_^_V 2 ^ = — + — . 

47T/A dt dx 

In the former case we derive at once 

at 

which merely expresses the fact that there can be no accumulation 
of volume density in the dielectric, on account of the assumed non- 
condensational character of the phenomena. 
In the latter case also 

V 2 F=0. 

Thus, when the dielectric has no initial charge, the charge is 
throughout confined to the interfaces separating media of different 
quality, such as the surfaces of conductors. 

It will form a sufficiently general case for our purpose if we 
consider the reflexion of a train of plane polarised waves at a plane 
metallic surface. The axis of z may be taken normal to the sur- 
face and the axis of x along it in train with the waves. If the 
vibration of the vector potential is in the plane of incidence, we 
have 

dF dH A _ n 

ax dz 

so that the variables are reduced to dependence on a single 
function % by the substitution 

dz ' dx 

The important case is when everything is periodic, and so 
involves the factor exp (— ipt). We have then 

tp ax ip dz 

and % is determined by the equations 

1 TK.,**! 



in the dielectric, and 



in the conductor. 



^K^ X df 



4>1Tfl A dt 



1891.] vibrating electrical system, and its radiation. 173 

At the interface F is continuous, so that -^ is continuous ; 

and H is continuous, so that 

dV 1 dV . 

r^ 7 — is continuous ; 

ax tp dz 

these follow from the definition . of F, 0, H as the potentials of 

volume distributions. 

Further, the component magnetic induction along the normal 

must be continuous across the interface, which being zero it is ; 

and the magnetic force along the interface must be continuous, 

,, x 1 (dF dH\ . in,. 

so that - -, j— is continuous across the suriace, that is 

ix \dz doc J ' 

— V 2 ^' is continuous, 

r' 

or by the equations for ■%', 

K d*Xi = izf d X* 

1 di z <r 2 dt ' 

or finally, for this special case of harmonic waves 

- l P K iXi = — Xv 

Let 
%/ = J.j exp i (Ix — n x z — pt) + B l exp i (Ix + n x z —pi), 

Xl = A 2 ex P l ( lx ~ n 2 z ~ ft)* 
A t , B 1 thus representing the coefficients of the incident and 
reflected waves, and A 2 that of the surface- wave in the conductor 
for which n 2 must in the result be complex. 

The value of I must be the same for all these waves, as 
their traces on the surface must move along it with the same 
velocity. 

The differential equations satisfied by ■% give 

^(l 2 + <)=/, 






<7 2 + n *) = tp. 



The first of these gives the velocity of the dielectric wave 
to be QiJZ^f , as it ought to be. The second gives the pene- 
tration of the surface-wave into the conductor by the equation 



±74 Mr Larmor, On a mechanical representation of a [May 4, 

If i denote the angle of incidence of the waves, \ their length, 
and v their velocity, we must have 

In + n x z — pt = — (ob sin i -f z cos i - vt). 
Thus P - ■ " ■ 



I 2 sin i 2tt sin i ' 

where v = {fi 1 K 1 )~* is of the order 3.10 10 C.G.S., and for light waves 
X is of the order 10~ 4 , and for copper a 2 is about 1600. 
Hence for all realisable wave-lengths, long or short, 



which for light waves is of the order 10 5 , and is very large for 
all realisable waves, unless for media of slight conductivity com- 
pared with metals. The amplitude of the surface wave in the 
conductor is reduced in the ratio e~ l at a depth n~ x ; this wave 
is therefore absolutely superficial, and is fully developed even 
in a very thin sheet of metal. 
The surface conditions give 

n 1 (A 1 -B 1 ) = n 2 A 2 , 

l(A 1 + B t -A 2 ) = - l -4m<r 
V 

-cpK^+BJ^A^ 

2 

of which the first and third equations determine the amplitudes 
of the waves produced by the reflexion of A lt and the second 
determines the surface density 

cr exp i {lx—pt) 

of the superficial electric charge. 

n„ 47r 



Hence p + ^ l) A 2 = 2A X , 



i.e. 



a„=2 ,vr»fi , A 



nj \pa- 2 K 1 

n 
and B=A--*A C 

1 x ?2j 



1891.] vibrating electrical system, and its radiation. 175 

The relative magnitudes of the two terms in the numerator 
of A 2 have to be estimated ; the terms to be compared are of the 
orders 

uAA 2 , X Xv 

— , and w or — , 

that is WX h and 10 7 \, roughly. 

For light waves, X is of the order 10~ 4 , so that neither of these 
terms can be neglected compared with the other ; and the com- 
pletion of the solution will correspond to the somewhat com- 
plicated circumstances of the metallic reflexion of light. 

But for waves comparable to a centimetre in length, or longer, 
the second term is negligible ; and then 



^- s sr^' 



and B 1 = — A 1 . 

The wave is therefore reflected clean but with opposite phase. 
And the value of n 2 given above shews that the longer the waves 
the slighter is their penetration into the conductor ; so that even 
for a curved surface like that of a wire this solution has an 
application. 

The meaning of this approximation is that the first surface 
condition, in the form it assumes when n 2 is very great, supplies 
all the necessary data for the motion in the dielectric. That 
surface condition is equivalent to the statement that F is zero 
in the dielectric along the surface, and therefore so is the tan- 
gential displacement. 

Thus the tangential surface conditions suffice in this case to 
give a full account of the dielectric phenomena, the normal con- 
ditions being simply left to take care of themselves. 

The function V by which the adjustment is made in the 
conductor to obtain the normal displacement at the surface which 
shall satisfy the condition of zero condensation is derived from 
the characteristic equation V 2 V=0, the same equation as that 
for the pressure in a homogeneous massless fluid ; it of course 
indicates instantaneous adjustment to an equilibrium value 
throughout the volume. 

(3) On the Theory of Discontinuous Fluid Motions in two 
dimensions. By A. E. H. Love, M.A., St John's College. 

This paper contains an exposition of a modification of Mr 
Michell's method published in Phil. Trans. R. 8., A. 1890. The 
motion of the fluid is supposed to take place in the plane of a 
complex variable z, and to be given by means of a velocity-potential 



176 Mr Love, On the Theory of Discontinuous [May 4, 

<£ and a stream-function i/r, so that <£ + tyjr or w is a function of #. 
The object of the theory is to show how in any given problem the 
functional relation between w and z can be discovered. For this 
purpose we consider two functions £ and fl such that 

£= dz/dw and II = log f, 

and it is well known that O is a complex quantity, whose real part 
is the logarithm of the reciprocal of the velocity of the fluid at the 
point z, and whose imaginary part is the angle which the direction 
of this velocity makes with the real axis in the z plane. In 
the kind of problems to which the method is applicable the region 
of the z plane within which the motion takes place is bounded 
partly by fixed straight lines and partly by free stream-lines. 
Along the fixed boundaries the direction of the velocity is given so 
that the corresponding parts of the boundary in the fl plane are 
lines parallel to the real axis. Along the free stream-lines the 
velocity is a given constant/so that the corresponding parts of the 
boundary in the 12 plane are parts of a straight line parallel to the 
imaginary axis. Hence the boundary in the fl plane is a polygon 
which we know how to draw. In like manner the boundary in the 
w plane, consisting of parts of straight lines parallel to the real 
axis {-^r = const.,) is a polygon which we know how to draw. If now 
we take an auxiliary complex variable u the polygons in the 12 
and w planes can be conformably represented on the half-plane for 
which the imaginary part of u is positive. The required trans- 
formations are given by the theory of Schwarz and Christoffel, and 
we are thus in a position to write down two relations 

dw j. , N 
dO, „ , . 

from which ^ = Gf^u) e$ f * ^ du . 

The roots and poles of the function f t (u) are values arbitrarily 
assumed to correspond to the corners of the polygon in the w 
plane, and the roots, poles and critical points of the function f 2 (u) 
are in like manner values arbitrarily assumed to correspond to the 
corners of tbe polygon in the O plane. Of these values three may 
be arbitrarily fixed, then the rest can be determined. The deter- 
mination is made by integrating the equation connecting z and u. 
If the limits of integration correspond to two critical points of the 
function 12 which lie on the part of the u line that corresponds to 
a fixed boundary we shall obtain an expression in terms of our 
assumed arbitrary constants for one of the dimensions of the fixed 



1891.] Fluid Motions in two dimensions. 177 

boundary. In this way there will arise sufficient relations to 
determine all the arbitrary constants. If we integrate the equation 
connecting z and u for values of u that correspond to a free stream- 
line we shall find for z a complex expression, so that the co- 
ordinates x and y of any point on the free stream-line will be 
given functions of a real parameter u. 

Mr Michell's theory rests on the properties of the function 
which is the real part of XI. He shows how to determine this 
function in terms of u by considering an analogous electrical 
problem. When it is known he deduces from it the differential 
relation between z and u that would be found by the method of 
this paper. 

After explaining the method I give solutions of the following 
problems : 

(i) Mr Michell's problem of the escape of a jet from a tank. 

(ii) The flow of liquid against a disc with an elevated rim. 

(iii) The impact of a jet against a finite lamina. 

(iv) The resistance offered by a plane obstacle placed in a 
canal of finite width. 

(v) The flow of liquid past a pier projecting obliquely. 

1. The theory of discontinuous fluid motions in two dimensions, 
as at present developed, rests essentially on two particular pro- 
positions to which we proceed. 

Prop. I. The motion being supposed to take place in the 
plane of the complex variable z, and to be given by means of a 
velocity-potential <£ or a stream-function ty, so that 

w = <J> + iylr=f (x + iy) =f(z), 

the quantity dz/dw, which we call £, is a complex variable whose 
modulus is the reciprocal of the velocity, and whose argument is 
the direction of the velocity of the fluid at the point z ; and the 
quantity log £, which we call fl, is a complex variable whose 
real part is the logarithm of the reciprocal of the velocity and 
imaginary part is V( — 1) multiplied by the angle the direction of 
the velocity makes with the real axis in the z plane. 

To prove this observe that if u, v be the velocities at any point 
parallel to x, y, then 

dw dw dd> d-fr 

dz ox ox ox 
,i „ dz u + iv u + iv 

SOthat ^ = dw = u^7 == ~^- S ^ 

and fl = logr=log^ 2 -=logi + ^ (A), 

where $ is such that cos 6 = u/q and sm0 = v/q. 



178 Mr Love, On the Theory of Discontinuous [May 4, 

Peop. II. It is possible to find a transformation by means of 
a relation between two complex variables Z and Z' by which any 
given polygon in the Z' plane can be transformed into the real 
axis in the Z plane, and points within the polygon in the Z' plane 
correspond singly to points in the Z plane whose imaginary part is 
positive. 

This transformation and the conformable representation of the 
polygon upon the half-plane, which it involves, have been in- 
vestigated by Schwarz and Christoffel, and it can be shown that 
the relation between Z and Z' is given by the equation 

%=AIi{Z-X r f^- 1 (B), 

where A is a constant and X r is the point on the real axis in the 
Z plane which corresponds to the internal angle a r of the polygon 
in the Z' plane. 

To verify this observe, 

(i) That dZ'/dZ is never zero or infinite except at points on 
the real axis in the Z plane : 

(ii) That if Z be real, and lie between two consecutive zeros or 
infinities of the function dZ'/dZ, say X r and X r+l , the argument of 
dZ'jdZ or dZ'jdX remains the same for all the values of X, so that 
the argument of dZ' remains the same, and all the points Z' which 
correspond to points on the real axis between X r and X r+1 lie in 
one straight line in the Z' plane. 

By combining (i) and (ii) it appears that the points on one side 
of the real axis in the Z plane correspond to points within a 
polygon in the Z' plane, and the points X x , X 2 , ... correspond to 
the corners. 

We must further observe (iii) that if Z be very near to X r all 
the other factors on the right of (B) may be considered constant 

except (Z-X r ) a ^ ir ~ 1 , so that in the neighbourhood of X r the 
change of the argument of dZ'jdZ is the same as that of this 
factor — and, in passing through X r in the positive direction, this 
change is an increase by it — a r , which is the same as the increase 
of argument of dZ' in going round a corner of the polygon where 
the internal angle is a r . 

We note that it is in general possible in a given problem to 
choose arbitrarily three of the points X v X 2 , ... "and then the rest 
will be determined. 

2. The problems to which the theory is applicable are such 
as are concerned with the motion of fluid in space bounded 
partly by fixed plane rigid walls and partly by one or more 
free stream-lines along which the velocity is constant. These 
include the escape of a jet from a polygonal vessel, flow into 



1891.] Fluid Motions in two dimensions. 179 

or out of straight tubes, flow past one or more plane laminas 
fixed in a finite stream whose boundaries are either free or 
fixed, or in an infinite stream. In all such cases there is a 
certain number of bounding stream-lines so that, with the no- 
tation of Prop. I., there will be a certain region in the z plane 
within which the motion takes place, bounded partly by given 
fixed straight lines and partly by unknown free stream-lines, 
and a certain corresponding region in the w plane bounded by 
parts of straight lines ^r = const. If the motion and the form 
of the free stream-lines were known we should know the relation 
between w and z which effects a conformable representation of 
the region in the w plane upon the region in the z plane in 
which the motion takes place. Conversely if this relation can 
be found we shall know the motion and the form of the free 
stream-lines. 

This relation can be found indirectly by the aid of the 
function fl introduced in Prop. I. Along a plane rigid wall 
the direction of the velocity is given, and along a free stream- 
line the velocity is constant. Hence along a plane rigid wall 
the imaginary part of 12 is a given constant, and along the free 
stream-lines the real part of 12 is a given constant. There will 
exist a relation between O and z by which could be effected 
a conformable representation of a certain region in the O plane 
upon the region of the z plane within which the motion takes 
place. The region in the O plane is bounded by parts of straight 
lines parallel to the axis of 12 real, corresponding to the plane 
rigid walls, and parts of a straight line parallel to the axis of 
£1 imaginary, corresponding to the free stream- lines. The 
velocity vanishes at angles of the rigid boundary in the z plane, 
and at points where stream-lines divide ; the corresponding points 
of the fl plane lie at oo in the direction of XI real, so that the 
general form of the boundary in the fl plane is as in the figure, 



where the thick lines correspond to rigid walls and the thin 
lines to free stream-lines, and there are as many thick lines as 
there are definite prescribed directions of motion. The inter- 
sections of thick and thin lines correspond to points where a 
free stream-line starts out from a rigid boundary. 

VOL. VII. PT. IV. 15 



180 Mr Love, On the Theory of Discontinuous [May 4, 

The relation between Q and z by which the representa- 
tion of the XI region upon the z region could be effected is 
unknown until the problem is solved, but, by applications of 
Schwarz's transformation given in Prop. TL, the XI region and 
the w region can each be represented upon the same half-plane 
in the plane of a new variable (u). In this way we can transform 
the X2 region into the w region and hence arises a relation 
£l=f(w) or log(dz/dw)=f(w). This is a differential equation 
defining z as a function of w, and when it is solved we shall 
know the region of z which is represented upon the region of 
w. Part of the boundary of this z region is prescribed and will 
inevitably agree with that obtained by solution of the differential 
equation, the determination of the other part will give the form 
of the free stream-lines. 

It is in general most convenient to take the constant velocity 
along the free stream-lines to be unity so that the corresponding 
value of the real part of X2 is zero, and to change the method 
of procedure sketched above by eliminating w and finding a 
differential equation between z and u. The region of the z plane 
within which the motion takes place will be that which by this 
relation is conformably represented upon the half-plane u. 

In fact we have a given polygonal boundary (formed of parts 
of parallel lines) in the w plane, and this can be transformed 
into the real axis in the u plane by a relation of the form 

dw , , . 

sir ■£<">■ 

And we have a given polygonal boundary in the O plane, and 
this can be transformed into the real axis in the u plane by a 
relation of the same form, say 

d£l , 

where the roots, poles and critical points of f 2 (u) are points 
u = a, ... assumed to correspond to the corners of the polygon in 
the X2 plane. Integrating this equation we obtain 

dz_ = Ce fU{u)du 
dw ' 

where G is a constant of integration. Hence we have the dif- 
ferential equation 

^- = Cf 1 (u)e fMu)du . 

OjUj 

If we integrate this for the parts of the line u real that lie 
between the critical points of the function X2, we shall find ex- 



1891.] 



Fluid Motions in two dimensions. 



181 



pressions for the lengths of the straight boundaries, which will 
determine the unknown constants a, and we shall find for z a 
complex expression when u has values corresponding to points 
on a free stream-line; i.e. the coordinates x and y of a point 
on a free stream-line are expressed as functions of a real para- 
meter u. 

All the problems solved in the first part of Mr Michell's paper 
can be treated in the manner here explained. I propose now 
to consider his first problem — that of the escape of a jet from a 
vessel,, and then to give solutions applicable to other cases. 

3. Problem (i). Escape of a jet from a vessel. 

Suppose we have a polygonal vessel of such shape that the 
fluid coming from oo must move parallel to the negative direction 
of the axis of y in the z plane, and the jet is formed by fluid 
escaping from a side parallel to the axis of x. 



a \ 




The boundary in the z plane will be as above, and as we are 
going to transform this boundary into a straight line u real in 
the plane of a new variable u it is convenient to denote points 
on the boundary by the values of u at the corresponding points. 
We shall assume then that the points u = l, u = — l correspond 
to the edges of the hole, and u = go to the point at go in the 
z plane in the direction of the jet, and we shall take u — c, 
a t , a 2 , ... unknown constants for the points corresponding to the 
point at co in the z plane from which the fluid comes and the 
corners of the polygonal boundary. 

The boundary in the iv plane consists of two infinite straight 
lines corresponding to the bounding stream-lines. If these be 
taken to be ty = 0, ^r — tt, and the velocity along the free stream- 
lines be taken as unity, the ultimate breadth of the jet will be ir. 

15—2 



182 Mr Love, On the Theory of Discontinuous [May 4, 

The point u = c will correspond to <£ = — oo , and u= cc to (f> = oo . 
The figure in the w plane is 

-l 



where the lower line is the axis of real quantities ^ = 0, and 
the upper is ^ = 7r. The strip between these lines can be con- 
formably represented upon a half-plane u by means of the trans- 
formation 

dw 1 



du u — c 
and the boundary in the u plane is 



.(1), 



— co — 1 e a x 1 oo 

4. To find the boundary in the Q. plane. 

Let a r be the value of u that corresponds to any angle a. 
of the polygon. The stream-line yjr = has for initial direction 
6 = — \tt. At the first angle a x the velocity vanishes, at oo it 
has a definite value. Thus log q' 1 increases from a certain value 
to + oo as u moves from c to a x , and 6 remains constant and 
equal to — \tt, and the corresponding part of the boundary in 
the H plane is therefore a straight line 6 = — \ir from a certain 
point for which the real part of XI is positive to oo . 

As u increases from a t to a 2 , log*/" 1 diminishes from oo to 
a certain minimum and then increases to oo , 6 remains positive 
and equal to — \ir + (tt — aj. Thus we have the two sides of 
a second line starting from an unknown point (say u = c 1 ) and 
running to oo in the O plane in the direction of O real. 

We shall obtain a line in the same way for each side of the 
vessel until we come to the side that contains the hole. On 
this line 6 = 0, and log q' 1 diminishes from oo to zero, its value 
at the edge of the hole where the velocity is unity. 

The next part of the boundary consists of the line log q' 1 = 
drawn from the origin downwards (because 6 becomes negative 
on the jet), to some value corresponding to w=». This corre- 
sponds to the free part of the stream-line aJt = 0. Passing through 
this point the line can be continued downwards to = — ir the 
value at the other edge of the hole corresponding to u = — 1, 
this part of the line corresponds to the free part of the stream- 
line ^r = 7r. 

Proceeding from this point we have to trace the line 6 = — ir 
to oo , then both sides of the other lines 6 = const, corresponding 



1891.] 



Fluid Motions in two dimensions. 



183 



to the further sides of the vessel, and finally the underside of 
the line 9 = — \ir from oo to the point corresponding to u = c. 
Hence the boundary is 

1 6i=0 



ft 00 



-1 











T V" "1/ 




o 












Jl 
T 

en 






6 = 


-i* 


Oj 












6C 

O 


















6 = 


— 7T 





where the point marked 1 is the origin in the fl plane and all 
points are marked by the corresponding values of u, and the 
figure is traced continuously by starting from any point, say c, 
going from c to a l} from a 1 to c x , from c x to a 2 , and so on, keeping 
the region marked out on the left. 

5. The H region can be conformably represented on the half- 
plane u by means of the relation 

u —c (u — c,) (u — O ... 

(2), 



dn = A 

du VC" 2- 1) (u — «j) (u — a 2 ) ... 



where the number of the letters a v a 2 , ... exceeds that of the 
letters c v c 2 , ..'. by two, and A is a constant which can be de- 
termined ; for the angles at 1 and — 1 in the H plane are each 
\tt, the angles at a v a 2 , ... are each 0, and the angles at c, c ,... 
are each 2tt. 

Now the function 

(u-c)(u-c,)(u-c 2 )... 
can be put into partial fractions in the form 



u — a. 



+ 



A, 



u — a„ 



+ 



where A lt A 2 ,... depend only on the c's and «'s, and thus we 
may write (2) in the form 



du 



= -AiX 



L(t»- O|i ).v(i-t0 



■(3), 



and the integral of this is 



n = -Ailog\B\J 



u — a 



where B is a constant which can be determined. 



A nlsJi l - "nh 

J 
(4), 



184 Mr Love, On the Theory of Discontinuous [May 4, 

Remembering that O = log (dz/dw) and using (1) we find 



dz 



B 



du u — c 



n 



'(1 - a n u) + V(l - O V(l - u 2 )' 



u — a m 



-Ai.AJJil-aJ) 



.(5). 



To determine the constants observe first that in passing the 
point a n the argument of u increases by ir — a n , so that the index 
— Ai . AJ\/(1 — aj) must be 1 — clJtt and the sum of these 
indices is unity. Also if u = 1 each factor in the product = 1 
and 12 = log B is zero so that B = 1. 



The differential equation for z thus becomes 



dz _ 1 
du u — c 



n 



]i-¥)W(i-Q#-4 



1 - ajir 



u-a n 
as in Mr Michell's paper, p. 401. 

We have also a series of relations of the form 

-Ai. A„, Ai (a n - c) (a n - c t ) (a n -c 2 )... 



(6), 



1 - — " = . 

7T V(l-0 



V(l-0 (a B -a 1 )(a w -a 9 ) 



.(7). 



There are as many of these as there are a's, that is, as many 
as the number of the c's + 1 and these would suffice to determine 
the c's and A in terms of the a's. 

In the particular problem worked out in detail by Mr Michell, 
viz. that of two rectangular corners a and b for which 

1 >b >c >a> — l, 

the above relations become 

Ai b — c _ Ai a — c _ 1 

~ J(l-b 2 ) F^ = ~V(1-^) o7^b~2' 

giving a relation 

(8), 



which determines c in terms of a and b, viz. 



c = 



vq-aO+vq-fr 2 ) 



• (9). 



1891.] Fluid Motions in two dimensions. 185 

This relation must hold among the constants of Mr Michell's 
problem. In his paper c is only determined in the case of 
symmetry for which it is obviously zero. 

I do not propose to proceed further with this problem at 
present. (See Art. 13 below.) 



6. Problem (ii). Liquid flowing against a disc with an elevated 



rim. 



Suppose that in an infinite mass of fluid moving from infinity 
in the negative direction of the axis of y there is a vessel bounded 
by three sides of a rectangle the missing side being parallel to 
the axis of x. The fluid will enter the vessel — one stream-line 
will divide at the middle point of the base, and there will be two 
free stream-lines starting out from the edges of the vessel, behind 
which there will be dead water. 





The boundary in the z plane will be as in the figure, and we 
shall suppose the points u = ± 1 to correspond to the edges of 
the vessel, u = ± a to the corners of the vessel, and u = to the 
point where the stream-line divides. 

The boundary in the w plane will consist of the two sides of 
the line ty = from the origin to + oo , thus 



oC 



-1 



This can be transformed into the real axis in the u plane by 



taking 



dw 1 , 1A v 

S=2" (10) ' 



and then the u boundary is 



186 Mr Love, On the Theory of Discontinuous [May 4, 

The £1 boundary is 



-1 







a 




61 = 




c 


0= -IT 





-c 


ft- 3,r 


-a 



where c and — c are the values of u at the points in the base of 
the vessel where the velocity is a maximum. 

7. This H region can be conformably represented upon the 
half-plane w by means of the relation 



dO _ A w 2 — c 2 

du \/(u 2 — 1) . u (w 2 — a 2 ) 



•(H). 



Writing this in partial fractions 
dO - Ai 



du V(l _ u *) 



VI a?-c 2 1 a 2 - c 2 1 
a 2 w 2a 2 w — a 2a 2 u + a 



and integrating we have 



n=log\B 



i + va-^y 

W 



ylic 2 



1 - aw + V(l ~ a 2 ) V(l - u *) 1 + «w + V(l - a 2 ) V(l - w 2 ) 
w — a tt + a 

so that 



2\-i-4i(a 2 -e 2 )- 



2oV(l-a s ) 



^ _ 1 R 

aw 2 



"i + v(i-u 2 y 



^ic 2 



l_ aM + ^/(l_ a 2 )V(l-^) l+aw+j/O-aVa-w 2 )! ^ ° 2) 



u — a 



u + a 



2a 2 V(i-« 2 ) 
.(12). 



Now as in the preceding, since the corners of the z polygon 
that correspond to 1 and — 1 are right angles 

-^(a 2 -c 2 ) _l 
2aV(l-<> ~2' 



1891.] Fluid Motions in favo dimensions. 187 

and since the z boundary has no singularity at a = 

Aic 2 _ 

Q^ Q 2 Q^ 

whence — 5 — = J(l — a 2 ) or c 2 = = ttz — —^- (13). 

c 2 v 1 + V(l - » ) 

Also £1 = + i— when w = l so that B = i, and the relation be- 
tween z and u is 

1 - au + V(l - a 2 ) V(l - w 2 ) 1 + au + V(l - a 2 ) V(l - -m 8 ) ' ' 
u — a u + a 

Now remembering the identity, 

V( l + a) \/(l + u) + V(l - a) V(l - u) 

we have 



= {1 + au + V(l - a 2 ) V(l - '^ 2 )} 2 > 



* - * [i ■+ V (i - o] V(1 -"?+^r J) (i4). 

du 2 L n *J{u> - a 2 ) ; v 7 

This corresponds to the case of liquid flowing directly against 
a disc with an elevated rim, the rim being the line in the z plane 
that joins the points corresponding to u = a and u = 1. Observe 
by way of verification that if we make a = 1 there will be no rim 
and we get the right transformation for fluid flowing against a 
plane disc, viz.: 

! = -l[l + V(W)] do). 

8. Suppose now that a is very nearly 1 and take 

l-a 2 = k\ d z = l-k 2 = k' 2 (16), 

where k is small. Then equation (14) may be written 

dz_i r V(l-Q , ,, V(l-a 2 ) 1-u* 

du~2 y{u 2 - a 2 ) [ X + Vii a ;i + V(w 2 - « 2 ) VO 2 - a 2 )} ' 
and the height of the rim is 



1 r \f-y) (1 + V(1 _ „*» + f^L + a^jI 



du 
..(17). 



188 Mr Love, On the Theory of Discontinuous [May 4, 

f 1 V(l - if) , _ f 1 (1 - ^ 2 ) du 
JaVfa'-a*) J a V(l-^ 2 )V(^-a 2 )' 

putting m = a sec (f>, the latter integral becomes 

•cos- 1 a (1 _ a 2 sec 2 <£) a sec <£ tan $d$ _ foos-^a (l - a 2 sec 2 <£) d<£ 
o a V(l — a 2 sec" <£) tan (/> J V(l — <*>* — sin 2 0) ' 

putting sin <£ = V(l — & 2 ) sin ^ = k sin -^r, we have 

7T 

2 F cos 2 ^ c£l/r 



'o [l-Fsin*^]* 

7T . . 

or h — in the limit when k is small. 

4 

f 1 du /"cos -1 a 

AgaiD 'i«VK^?) = io SeC * # 

= k in the limit 

fi (l— u 2 ) roos- 1 a ' 

and -^-= tgr du=\ s (1 - a sec 2 <6) sec d>arf> 

JaV{u -a) Jo 

/"cos -1 a (X 2 f -l Tcos -1 a 

= .1 sec <£ cty - -s- j [sec <f> tan <£] COS a + I sec <£ (20 

1+F, /1+fc 1 7 

= as far as & 2 . 
Thus neglecting F the height of the rim is 

To find the breadth of the disc we must write (14) in the form 

^=2 [1 + V(1 " W)] V(1-^ 2 -^ 2 )' 

and integrate from u = to u = a and double the result. We may 
reject k altogether, and thus find the breadth 

a + \ [sin -1 a + a */(l — & 2 )]> 

or in the limit when a is very nearly 1 we may take the breadth 
equal to - 

^ (18). 

thus the ratio height of rim to breadth of disc = ^k 2 = e say. 



1891.J 



Fluid Motions in two dimensions. 



189 



When u > 1, we must write (14) in the form 



£-£n-.v<tf-i>] 



-|a+*) 



ft-tV(tt g -l) 
V(w 2 -1 + F) 

1) 



V(w 2 -l+-t 2 ) + 2 



U 9 -l 



y (i* a - 1 + &■) VO 2 - 1 + & 2 )_ 

For the imaginary part of £ taking cosh = ujk' we have 



-■•|y(0 + sinh0cosh0)-0 



+ const., 



or writing z = %+ ty 
1 

y=2 



(l + k)0- 5—^ (0 + sinh cosh 6) 

dy 



+ const. 



The greatest value is given at once by -j- = 0, so that 

l=Jc, 

Je + 1 1 



or 



u 

cosh 2 6 = 



l-F 1-k' 
giving sinh 6 = *Jk (1 + ^k), 

0=y&(i + p). 

The value corresponding to u = 1 is given by 

COSll 2 = - To , 

giving sinh = k(l + |& 2 ), 

= jfe(l + p 2 ). 

Hence the height above the rim through which the free stream- 
line rises before turning back is 

i(iH-*)v*a+i*)-iv*[(i+t*)+ a +i*)a+*)] 

- 1 (1 + k) k (1 + p 2 ) + ^ [(1 + iF) + (1 + p 2 ) (1 + ¥)}, 

and this is ultimately 

5 i$ 

rejecting higher powers of k. 

Hence if e be the ratio height of rim to breadth of disc the 
greatest height above the rim to which the free stream-line rises 
before turning back is 

^ (2e) f of the breadth (19), 



or about 



a e* of the breadth, 



190 



Mr Love, On the Theory of Discontinuous [May 4, 



e.g. if the rim be yg-th of the breadth the stream-line rises through 
about ^jth of the breadth. Except quite close to the disc 
where u l — 1 is small we may reject k altogether, and the form of 
the free stream-line at a distance from the disc is the same as if 
there were no rim. 

To find the pressure on the disc we have to find 

dz 



which is 



\p 



or \p 



hp 



= iP 



l— 



i-[ r *? 

dz 



u 2 (1-u 2 - F) 



du 



du, 



k + V(l — u 
~ 1(l (1 + cos e) 2 (k + cos e) 2 - sin 2 6 cos 2 e 



v^lg ti+va-v)]^ 



dd, rejecting & 2 , 



o 
sin -1 a 



sin -± a 



(1 + COS 6) (k + COS 8) 

[(1 + cos 6)(k + cos 0) - (1 - cos 0) cos 6{l-k sec 0)] d<9 
"(2 cos 2 9 + 2k) dd 



\p [sin -1 a 4- a y^l — a 2 ) + 2& sin x a] 



= p 



7T A; /C7T 

I + 2 + ~2~ 



to the first order in k. 



or 



The breadth of the disc, rejecting k 2 , is 

a + i [ srn_1 « + a V(l — °0] +k[a + sin" 1 a], 
1 + -7- + - + &+-& to the first order in k. 



Thus the mean pressure, when the velocity on the free stream- 
lines is V, is 



>F 



7T 7 8 + 4tt + 2tt 2 
+ A; 



4 + 7T 



(4 + 7T) 2 



or 



Trp 



F 2 



4 + 7T 



■ 8 + 4tt + 2tt 2 .. ' 
■ 1 + (4 + 7r) a V(26) 



.(20). 



Comparing this with the case where there is no rim we see that 
the mean pressure is increased by about f ye of itself, e being the 
ratio height of rim to breadth of disc. 



1891.] 



Fluid Motions in two dimensions. 



191 



9. Peoblem (iii). Oblique impact of a jet upon a finite lamina. 

This problem includes Mr Michell's of the impact of a jet 
against an infinite plane, and also the problem of the impact of an 
infinite stream against a finite lamina worked out by Kirchhoff 
and Lord Rayleigh. 

The ^-region is bounded by two free stream-lines, by the lamina, 
and by the continuations of the stream-line that divides on the 
lamina. 




We shall suppose that u = ± 1 correspond to the ends of the 
lamina, and u = a to the point where the stream-line divides, u = b 
and u = c to the points to which the two streams go. 

The w-region will be bounded by the two infinite lines yjr = 
and yjr = tt say, making the breadth of the original jet ir, and by 
the two sides of a line ty = (3 say, where it > (3 > ; thus 



a\Z 



-1 



and this region can be conformably represented upon the half- 
plane u by means of the relation 



dw 
du 



u — a 



(u — b) (u — c)' 



•(21), 



and then 



"dr = 



a—b 
~c~^b 



192 



Mr Love, On the Theory of Discontinuous [May 4, 



will be the stream-line that divides, and the boundary in the 
u plane is 



-1 



The boundary in the fl plane is easily seen to be 
1 0=0 



a 



-i 



This O-region can be conformably represented upon the half-plane 
u by means of the relation 

(21), 



dO L _ A 

du \f{u 2 —l)u — a 



Ai 



whence O = log 



l-aw+V(l-a 2 )V(l-w 2 )' 



so that -7- 
du 



u — a 
l-au + y/(l-a?)^(l-u*) 



v/(l-a 2 ) 



in which the index — 



(u — b) (u — c) 
Ai 



•(22), 



has been determined to be unity by 



V(l-0 

considering that there is no singularity in the z boundary at the 
point corresponding to u = a. 

10. We may now suppose that the lamina is part of the axis 
of %. If we take u to lie between — 1 and 1 and integrate (22) 
between these limits we shall get an expression for the breadth of 
the lamina. 

Writing the equation (22) in the form 

dz ab — 1 1 ac — 1 1 
du~ c—bu — b c — b c — u 



+ 



V(l-a 2 ) V(l-a 2 ) 
V(1-m 2 ) c-b 



b 2 -l 



+ 



c 2 -l 



_(u - b) V(l - u 2 ) (0 - u) V(l - u\ 



1891.] Fluid Motions in two dimensions. 193 

we find for the breadth of the lamina 
ac — 1 , c — 1 ab — 1 , 1 — 6 



c-6 ° c + 1 c — b & -1-6 

+ V(l - a 2 ) 7T - V( c 1 J 6 a2) W(¥ - 1) + V(C 2 " 1)] 7T. . .(23). 



•(24), 



When w is > 1 we must write (22) in the form 

dz _ an - 1 + i V(l - a 2 ) V(^ 2 - 1) 
du (u — b)(u — c) 

where a = cos a and ^(1 — a 2 ) = sin a, a being the angle the im- 
pinging jet makes with the lamina. The sign of the imaginary 
term is determined by considering that when u = go the argument 
of dz is a. 

When u lies between 1 and c we have 

dz _ ab-1 1 ac - 1 1 . V(l - a 2 ) 
du c — bu — b c — bc — u »J{u 2 — 1) 

V (l - a 2 ) [b 2 -l c 2 - 1" 



(c — b) */(u 2 —l) \_U — b C — U_ 

and the argument of dz in the neighbourhood of u = c is the same 
as that of 

ac — 1 . \/(l — a 2 ) 



c-b V(c 2 -l>(e-6)' 

or of - (ac - 1) - % V(l - 0/V(c 2 - 1). 

Thus the ultimate direction of this part of the stream is 0, where 

R cos 6=1 —ac, 

Rsmd = - V(l - a s )/V(c 8 - 1), 
giving R = c — a, 

and tanfl = - n V( \~? n (2-5). 

(1 — ac) v(c — 1) 

In like manner the ultimate direction of the other part may be 
determined. 

11. The form of the free stream-line from 1 to c would be 
found by integrating (24). I shall suppose u = + 1 to correspond 



194 



Mr Love, On the Theory of Discontinuous [May 4, 



to z = 0, and then the co-ordinates of a point on the free stream- 
line are 



ac — 1 , c 



u ab — 1 , u — b 



c — b " & c — 1 
y = Aj(l — a 2 ) cosh -1 it — 



c-b 

V(l - a 



log 



1-6 



c-b 



"^pzD^-M^.D^-DI 



V 



u — b 



y (26), 



+ V(c - 1) smh 1 — *-^ 



c — u 



in which w is a real parameter lying between 1 and c. 

To find the free stream-lines that bound the jet, we have to 
integrate (24) up to values of u lying beyond u — c. We choose 
the path of integration to start from u = 1 and proceed along u 
real very nearly to u = c, then over a little semicircle whose centre 
is u = c and then again along u real ; we thus find 

ab — 1 , u 

=-t^ 1o «t 



z = 



b ac 
-7- + 



1, u — c 

— r- log — 

e — 6 ° c — 1 



i\ -u-i .\/(l-a 2 ) 



-V(o'-l)smh-j^^lM^l) 

I 11 C J 



• qc- 1 . V(l-a ! ')V(c'-iy 
c — 6 c — 6 



'y^-Dsinh-j^-p^'- 1 ) 



where the last line is the part contributed to the integral by the 
small semicircle. Hence on this stream-line 



ac — 1 



x = - — r log - 
c—b ° c— 1 



— c at — 1 



o-6 



log 



-6. 7rV(l-OV(c 2 ~l) 



1-6 



+ 



c-b 



y = V(l - a 2 ) COsh" 1 U - ^L_|5 



^-Dsinh-f^-y^- 1 )! 



u — b 



^-D^ ^-^r- 1 ^ ]- 



ac 



(27). 



c — 6 , 



This stream-line will have an asymptote x = y cot a + oc which is 
to be determined by making u infinite and 



a = cos a, V(l _ °0 = sm a - 



1891.] Fluid Motions in two dimensions. 195 

When u is very great, we can expand x and y in the forms 
x= a\ogu + A n -\ -+ ... 

o o u 

and, to find the asymptote, we must find A and B Q . We get 



x— a log u + 



y = V(1- a 2 ) log u + 



^zIlog(l-6)-^-ilog( C -l) 

c-b oV o-6 oV ' 



+! r_ya-«V(c 2 -i) 



c-6 



. 1 

+ terms in - , 

u 



V(l - a 2 ) log 2 



- ^L_^2 {^(6* _ i) cosh-^- 6) - V(c 2 - 1) cosh^c] - tt^— =J 

+ terms in - , 

u 

so that 

# n = — r-r- log (1-6) r- log (c — 1) H LA — \ y 

c-6 &v 7 c-6 oV y c— 6 

- a log 2 - -^=- [v / (6 2 - 1) cosh" 1 (- 6) - ^(c 2 - 1) cosh _1 c) 

C 

air ac-l 
+ V(l-a") o-6 (28) ' 

and x — ^tt cosec a is the distance to the right of the right-hand 
end of the lamina of the point where the initial middle line of 
the jet strikes the plane of the lamina. This is the point marked 
P in the z figure. 

There are now sufficient relations to determine the unknown 
constants a, 6, c. A jet of given breadth coming from oo in a 
given direction strikes the lamina obliquely in such a way that 
the middle line of the jet passes through a certain point in the 
lamina. We choose the unit of length so that the given breadth 
of the jet may be it. Then the constant a is determined by 
making a = cos a, *J(1 — a' 2 ) = sin a, where a is the angle between 
the direction of the impinging jet and the lamina; and there are 
two relations to determine the constants 6 and c, viz., equation 
(23) gives the breadth of the lamina in terms of 6 and c, and 
equation (28) enables us to identify a point P whose distance from 
the right of (1) is given with a point whose distance from the 
right of (1) is a given function of 6 and c. 

VOL. VII. pt. iv. 16 



196 



Mr Love, On the Theory of Discontinuous [May 4, 



The particular cases investigated by Kirchhoff and Lord Ray- 
leigh include all that have any practical interest at all comparable 
with the analytical difficulties. 

12. Problem (iv). Plane obstacle situated in canal of finite 
width. 

A stream flowing between two fixed parallel planes impinges 
upon a lamina fixed across the stream at a given angle a. 

The ^-region is bounded by the two fixed planes, the lamina, 
and the continuations beyond its edges of the stream-lines that 




divide upon the lamina. We shall suppose the point at oo from 
which the stream comes to correspond to u=<x>; the points to 
which its two parts go, to u = b, c; the point where the stream- 
line divides, to u = a ; and the edges of the lamina to u = ± 1. 
Then we must have 

c> 1 > a > — 1 > b. 

The w and u regions will be precisely the same as in the 
last problem, and the relation between them will be 

dw_ u — a , 2 „ 

du (u—b)(u — c) 

The fl boundary is easily seen to be 
l 6>=0 



n 



e^ -a 



'A* 



1891.] 



Fluid Motions in two dimensions. 



197 



where the point marked oo corresponds to the point from which 
the stream comes and the point marked 1 is the origin in the 
fl j)lane. 

Now this polygon has right angles at 1, — 1, c, b, an angle 
at a and an angle lir at oo and it can be conformably represented 
upon the half-plane u by means of the relation 

d J} = A (30 ) 

In the general case the integration will require elliptic 
functions and I do not propose to proceed with it. 

13. In the case of symmetry it is clear that there is no 
flow across the line perpendicular to the lamina at its middle 
point so that this line may be treated as a real boundary and 
the a-reo-ion will be 



and the image of this in the line A B. It is clear that the 
motion takes place as if the boundary were ABODE. If now 
we reflect this in the line DE we get the same figure as in the 
escape of a jet from a rectangular vessel by an orifice in the 
middle of the base. This is a particular case of our first problem 
and the form of the jet has been worked out by Mr Michell. 
The relations between z, w, and u, are 



dw 
du 
dz 
du 



l v(i-« 2 ) + V(i-0 



(31), 



J(u 2 -a 2 ) J 

and we propose to find the pressure on the side of the vessel 
in which the aperture is. If we take the part to the left of 
the aperture we shall have for the difference of pressure on the 
two sides of this plane 



J a 



-SX 



dz 

.- du 
du 



.(32) 



1G— 2 



198 Mr Love, On the Theory of Discontinuous [May 4, 

where p is the density of the fluid, or 

u 2 — a 2 



h 



l- 



2 - d l - u 2 + 2 v /(l - a 2 ) V(l - u 2 )_ 

V(l-g 2 )+V(l-^) du 
\Ju 2 — a 2 u 

f 1 { V(l - u 2 ) + V(l - a 2 )} V(l - u 2 ) du 
p J a y(l-a 2 ) + ^/(l-u 2 )}^(u 2 -d 2 )- u ' 



or 



V(i-m'0 ck 



a V(w 2 — tt 2 ) w 
The value of the integral is 

7T/1 



2 W 1 

Returning now to the problem of the lamina we see that 
the pressure on it is 

pir(l-a)/a (33), 

and in the same case the velocity of the stream at infinity in 
the direction from which it comes is 

o/{l + V(l - a 2 )} (34), 

the breadth of the stream is 

d = 7T [1 + V(l - a 2 )}/a (35), 

and the breadth of the lamina is 

7 1 - a V(l - « 2 ) / o ■ -i x M m 

l = 7r + — '-{it — 2 sin l a) (36). 

Now let the stream flow from oo with a given velocity V in 
a canal of breadth d, and impinge symmetrically and directly on 
a pier of breadth I. Then if a quantity a be determined from the 
equation 

l 1 - a + V(l - a 2 ) (l - | sin -1 a] 

d = ~ V(l - a 2 ) ' 

the pressure on the pier will be 

n _ (1-g ) {1 + Va- a 2 )} 2 



a 2 [tt (1 - a) + V(l - a 2 ) (tt - 2 sin" 1 a)} ' 
By writing a = cos a we find the convenient form 



1891.] Fluid Motions in two dimensions. 199 

and the mean pressure is 

pirV 2 (suc 2 a — sec a) (1 + sin a) 2 



7r (1 — cos a) + 2a sin a 
where a is determined by (37). 



.(38), 



14. When the sides of the canal are distant, I is small com- 
pared with d, and if we take Ijd = e we shall have 



so that 



2 « 1 1 3 

a= ^fi 6 -^(^T4) eSnearl y (40) - 



As a first approximation taking a small, we have for the mean 
pressure 

pirV*($a*) _PttV 2 



a 2 , 4+7T 
7r 2- +2a 



•(41), 



the same as for a lamina held in an infinite stream. Going to a 
second approximation we have merely to retain the term 2a in 
the expansion of (1 + sin a) 2 and this gives for the mean pressure 



PttV 2 

4 + 7T 



4-7T 

4 + 7T 



•(42), 



so that the effect of the sides of the canal is to increase the mean 
pressure by 

4 P 7r 2 F 2 



(4 + tt) 2 



.(43), 



and the fraction of itself by which the mean pressure is increased 
is about \e, in which it is to be remembered that e is the ratio, 
breadth of pier to breadth of canal. 



15. Problem (v). Stream flowing past an obliquely projecting 
pier. 

Suppose the z boundary consists of parts of two straight lines 
one of them infinite in one direction and terminated at the point 
marked 0, and the other finite and inclined to the first at a given 
angle ir — a, and that the fluid is on the side of the boundary 



200 



Mr Love, On the Theory of Discontinuous (May 4, 



within the angle ir — a and comes from oo in the direction indi- 
cated. The figure in the z plane is 




There being but one bounding stream -line, the figure in the w 
plane is 



so that there is no necessity to transform to a new u plane. We 
take then the point w = to be the corner, w = 1 the extremity 
of the broken line and w = oo the point to which the stream goes. 

The figure in the O plane is 



= 



n 



>= -(w-a) 



and this can be conformably represented on the half-plane w by 
means of the relation 



d£l_ A 

dw V( w — V).w 



.(44), 



1891.] 



Fluid Motions in two dimensions. 



201 



so that 



XI = loo- \B 



'i + V(i-«0" 



Ai 



= log 



dz 
dw' 



- V(l - w) 

and since II = when w = 1, and the argument of dz increases by 
(it — a) as w goes through z%xo, we find 



dz 
dw 



l + vXi-w) 

1 - V(l - w) 



1 — a/7r 



.(45). 



16. With our choice of constants the length of the part 
between the points corresponding to w = and w = 1 is 



J" 

Jo 



1 + V(l - wj 
1 - V(l - w). 



1 - ajir 



dw, 



putting w = sin 2 and tan \6 = x, ajir = n, we transform this into 



(I-* 2 ) 



dx. 



.(46), 



o (1+tf 2 ) 3 

and when a is an exact submultiple of tt, n is an integer and we 
shall be able to evaluate the integral. 

The pressure on the part between the same two points is 



J o 



dw 
dz 



dz 
dw 



dw, 



or, making the same transformations as before, 



4/j 



^ 



') (1 -<*) s t 

n o (X-vb 



.(47). 



o (l+<> 3 

The above might be applied to find the pressure on the rudder 
of a ship when turned obliquely to the length of the ship. The 
average pressure, the centre of pressure, and the moment of the 
fluid pressures can all be expressed by means of definite integrals 
of similar form to the above. If it were worth while tables might 
be constructed giving the values of these quantities for any given 
inclination of the plane of the rudder to the longitudinal plane of 
the ship. As however the motion here considered is in two 
dimensions it is unlikely that the formulge would yield any result 
of use in Navigation. 



(4) On thin rotating isotropic disks. 
Fellow of King's College. 



By C. Chree, M.A., 



The following solution might be shortened by assuming certain 
results from a previous paper in the Transactions 1 . As the 
subject appears, however, to be one of considerable practical 

1 Vol. xiv. p. 328 et seq. 



202 



Mr Clvree, On thin rotating isotropic disks. [May 4, 



importance, I have on the advice of Professor Pearson made the 
proof complete in itself. 

The term disk is here restricted to mean a thin plate of which 
a section parallel to the faces is bounded by a circle or two con- 
centric circles. The disk is supposed of uniform density p and 
of an isotropic material, for which m and n are the elastic con- 
stants in the notation of Thomson and Tait's Natural Philosophy. 

Taking the axis of the disk for axis of z with the origin in 
the central plane — or plane bisecting the thickness of the disk — 
we see from the symmetry that as the disk rotates about its axis 
with uniform angular velocity co the displacement at every point 
is in the plane through that point and the axis, having for its 
components w parallel to the axis and u along the perpendicular 
r on the axis directed outwards. At any point r, z in the disk 
the strains are as follows : 



Normal strains. 



Tangential or shearing strain. 



dw 



longitudinal -j- parallel Oz, 
radial -=- along r, 



du dw , „ , „ 

T- + -p- m the plane ot zr. 
dz dr 



transverse - perpendicular to r and z. 

It is obvious from the symmetry that the two other tangential 
strains must vanish. 



The expression for the dilatation 8 is 

£ _ du u dw 
dr r dz 

The stress system is as follows : 



_ s dw 

iz = (pi — ft) 8 + 2ft -=— parallel to oz, 

Normal J ^ ^ u 

stresses | rr = (m - n) 8 + 2n ^- along r, 



.(1). 



4>(j> = (m — n)8+ 2n u/r perpendicular to Oz and to r ; 

(~ (du dw\ . , 

r =7i U + ^J inplane ^ 

The other two shearing stresses must vanish from the symmetry. 



Shearing (~ 
stress 



1891.] Mr Ghree, On thin rotating isotropic disks. 203 

We may suppose the disk at rest, acted on by a " centrifugal 
force" (o 2 pr per unit of volume. Thus the internal equations are 
the following two 1 

drr drz . rr — <p<p . 2 a /n\ 

-7- + -7-H — +et>pr = (2), 

dr dz r ' 

dp + 7; + d£ =0 

dr r dz v ' 

Let 21 denote the thickness of the disk, a the radius of its outer 
a' that of its inner cylindrical surface, or edge. Then supposing 
the disk exposed to no surface forces, the solution ought to satisfy 
the following surface conditions 2 — 

fz = (4), 

over the flat faces z = + I, i~ _ /KX 

" l« = (5), 

over the edges r — a and r=a, J 



[rr = (7). 

Substituting for the stresses their expressions in terms of the 
strains and using (1), we easily transform (2) and (3) respectively 
into 

, N d8 d { (du dw\) „ „ 

(»+.), - + „ s | r ^ _ _j| =_ kV (8) , 

f . d8 d [ (du dw\) 

(m+n)r &-"di-\ r U-^rr (9) - 

Differentiating (8) with respect to r and (9) with respect to z, 
then adding and dividing out by (ra -t- n) r, we get 

dr 2 r dr dz 2 m + n 

Of this a particular solution is 

S = -o) 2 pr 2 ^2{m + )i). 

A complementary solution in ascending powers of r and z with 
arbitrary constants can be obtained, as I have shown in a previous 
paper 3 . Of this we require for our present purpose only the 
constant term and that of the second degree, or 

8 = A + C(z 2 -±r 2 ). 

1 Pearson's Elastical Researches of Barre de Saint-Venant, foot-note, p. 79, or 
Ibbetson's Mathematical Theory of Perfectly Elastic Solids, p. 239. 

2 Ibbetson's Mathematical Theory 1. c. , or Todhunter and Pearson's History 

of Elasticity, Vol. i. Art. 614. 

3 Transactions, Vol. xiv. p. 328 et seq. 



204 Mr Ghree, On thin rotating isotropic disks. [May 4, 

Putting the right-hand side of (10) zero, it is easily verified 
that this is a solution. We thus have 

8 = A + G (z 2 - if 2 ) - ift)Vr 2 /(m +n) (11). 

Noticing that 

d ( (du dw\] _ d<$ d 2 w d I dw^ 
dr\ [dz dr)} dz dz 2 dr\ dr , 
we may transform (9) into 

d 2 w 1 dw d 2 w m d8 2m ri ... _. 

t? + -j-+TT = 7T = Gz ( 12 )- 

dr r dr dz n dz n 

Of this a particular solution is 

lu = -™Cz\ 
on 

A complementary solution is easily obtained in ascending powers 
of r and z. Of this we require for our present purpose only the 
terms of the first and third degrees, which separately satisfy (12) 
when the right-hand side is zero. Thus we get 

07) 

w = 0L 1 z+e 1 (2z 3 -ozr 2 )- (> - Cz 3 (13), 

oil 

where a x and e x are new constants. 

Employing (1) and (11), we in like manner easily transform (8) 
into 

d 2 u 1 du u d 2 u _ m d8 <o 2 pr 

dr* r dr r 2 dz* n dr n ' 



A particular solution is 






= r[G ™-^P-) (14). 

n m + nj 



711 ft) p 



c- 



n in + n 



For the complementary solution we require only the terms in odd 
powers of r and z up to the third degree. Terms in negative 
powers of z are of course inadmissible, and r~ l is the only negative 
power of r which satisfies the differential equation. Thus the 
complementary solution is 

u = - + ar + e (z 3 - 2zr 2 ) + £ {^z 2 r - r 3 ), 

where D, a, e, £ are new constants whose coefficients separately 
satisfy (14) when its right-hand side is 0. Thus for our complete 
solution of (14) we get 

u = 5 + ar + e(z 3 - 2zr 2 ) + Wz 2 r -r 3 ) + l (c - - -^-) r 3 . . .(15). 
r 8 V n m + nj 



1891.] Mr GJiree, On thin rotating isotropic disks. 



205 



The constants in (11), (13) and (15) are not all arbitrary, being 
connected through the identity (1). From it we find 

a x = A- 2a, 

1 m + n n 4 

6 = 0. 

Thus the solution we have arrived at is 

8 = A + G (z 2 - Y) - !«> 7(m + n), \ 

w = (A-2*)z- g Cz> + I (^ G - 8?) (2* 3 - Szr 2 ), 



u = — + ar + £(4*V - r s ) + ~ f — C- 



o) p 

m + n 



.(16), 



- + ar+^(4>z 2 r-r 3 ) + - - 

where all the constants are independent. 

To determine the constants we have the surface conditions 
(4)-(7). 

From (4) and the expression for zz in terms of the strains we 
easily find 

(m + n)A- 4wa + I 2 {(m + n)C- 16»f } 

+ ^| 8 «C-l(3m + n) 67-1^^ = 0.. .(17). 

Since this holds for all values of r between a' and a, the constant 
part and the coefficient of r 2 must separately vanish. Thus we 
get 

(m + >i) 4 - 4na + Z 2 {(m + n) (7 - 16<} = (18), 

8nK-USm + n)G-~ co 2 o = (19). 

2 2 in + n r v ' 

From (5) and the expression for rz in terms of the strains we 
at once obtain 

16n£-(m + n)C=0 (20). 

The equations (18), (19) and (20) are satisfied by 



m + n . 

a = — A A, 

4m 



n \ in — n ,, 

O = — t ; CO'O, y 

2 m (m + n) 
y 1 m — n 2 



.(21). 



206 Mr Chree, On thin rotating isotropic disks. [May 4, 

Substituting these values in the expressions for the strains 
and stresses we find for all values of r and z 



9 = 0,\ 

du dw _ 
dz dr ' 



.(22). 



and so rz = , 

Since rz is everywhere zero, the condition (6) over the edges is 
exactly satisfied. The only surface condition left is (7), but this 
we cannot exactly satisfy, unless m = n, by means of the present 
solution. For, substituting the above values of the arbitrary con- 
stants, we obtain from the expression for rr in terms of the strains 

™V=» = i (3m — n) A — 2na~ 2 D 

7m — n„ „ (m — n)(3m — n) „ „ /nox 

^ m 2 pa 2 - v — = — p r— L co 2 pz* (23), 

16m r 4>m(m + n) r 

rr r=a ' = similar expression, replacing a by a' (24). 

It is obvious we cannot make these stresses vanish for all values 
of z. 

If the thickness 21 of the disk be of the same order of mag- 
nitude as the radius a this failure renders the present method 
inapplicable; but when l/a is small it is easy to obtain a solution 
which according to Saint- Venant and other eminent authorities 
must be very approximately exact except in the immediate neigh- 
bourhood of the edges. 

The principle this solution is based on is that of statically 
equivalent systems of loading. According to this principle when 
a surface of an elastic solid has a small dimension — such as the 
thickness of a thin disk — all systems of surface forces which in 
their distribution along the small dimension are statically equi- 
valent produce, except in the immediate neighbourhood of their 
points of application, practically identical strains and stresses. 
We may thus for practical purposes replace any system of surface 
forces over the small dimension by any statically equivalent 
system. 

For a discussion of this principle and illustrations of its 
application, the reader is referred to Saint- Venant's Theorie de 
L'Masticite...de Clebsch, p. 174 et seq. and p. 727 et seq., also to 
Pearson's Elastical Researches of Barre de Saint-Venant, Arts. 8 
and 9. 

In my previous treatment of this problem 1 , which was limited 
to a complete disk, I determined the constant A so as to make 

1 Transactions, Vol. xiv. pp. 334 — 5, § 76, first two cases ; and Quarterly Journal, 
Vol. xxiii. 1889, pp. 24—28. 



1891.] Mr Chree, On thin rotating isotropic disks. 207 

rr r=a vanish for a given value of z, viz. either z = or z = ± I. In 
either case we are left with a system of unequilibrated normal 
forces along each generator of the edge. On this ground Professor 
Pearson has recently 1 expressed his opinion that my solution 
cannot be regarded as " final " even for a thin disk. As the 
normal forces in question are of the order of the square of the 
thickness of the disk, I am not altogether sure what weight may 
be attached to this criticism. In deference however to Professor 
Pearson's opinion, and to what I believe the view Saint-Venant 
would have taken, I propose the following method of solution 
which removes at least this objection. 

It consists in determining A and D from the equations 
•+i 

ft r=a dz =0 (25), 

-+i 

£ r=a >dz = (26). 

■i 

This still leaves normal stresses of the order of the square of the 
thickness over the edges, but the forces along each generator of an 
edge form a system in statical equilibrium. Thus according to 
the principle of statically equivalent systems, the solution we 
shall obtain — which must be strictly limited to thin disks — gives 
expressions for the strains and stresses which can differ sensibly 
from those supplied by the complete solution only in the im- 
mediate neighbourhood of the edges. 

For the case of a complete disk D must vanish and A is to be 
determined by (25). 

For the annular disk we find from (25) and (26) 
. 7 m — n 2/2 «\ . m — n „ 7 A 

&m(3m — n) r bm(m + n) r 



T\ ' "'' ^ 2 2/2 

JJ = -7^ (o pa a 

Szmn 



.(27). 



Also from equations (21) we have G and £ determined explicitly, 
and a found in terms of A. Thus all the constants of our solution 
are determined. For a complete disk we have only to put D = 0, 
and a' = in the expression for A in (27). 

The physical results attainable from the solution will perhaps 
be rendered more practically serviceable by replacing the m, n of 
our previous work by Young's modulus E and Poisson's ratio v. 
To express the values obtained above for the arbitrary constants 
in terms of E and n we require the relations 

m = %E/{(l-2 v )(l+ v )} r 

1 Nature, 1891, p. 488. 



208 Mr Chree, On thin rotating isotropic disks. [May 4, 

Substituting the expressions found for the arbitrary constants in 
terms of E and 77 in (16), we find for the strains in an annular 
disk 

S = ^% m {(3 + i){a? + a 2 ) - 2 (1+ v ) r 2 } 

+ g VV-2V)( 1 + V) (P ._ 3/) (29 ) ; 

-^ 2 r^ ( * W) •••- (30) ' 

= g |(1 - *7>(-3 + ^(a 2 + a' 2 ) r - (1 - V 2 ) r 3 + ~ (1 + v)(o + ,)} 

+ ^(l +*,)?• (Z 2 -3* 2 ) (31)- 

From these strains we find for the stresses 
S = ^ /9 .4- ^ L 2 4- ffl * - •* _ ^1 



M 



(3 + 7/)ja 2 +a' 2 -r 



r 



+ ^^(Z 2 ~-V) (32), 

= -. + ^|(l_^)y + ( 3 + 97 )^i (33). 



For a complete disk we have 

4# 



8 = "V(l_2?) K3 + iy)a ._ 2(1 + iy) > 1 



+ 3^^ (1 "l- ( , 1+??) ^- 8 ^ < 84 >' 



W = -H^K3 + ^^-2(l + ^r^} 



-^ 2 i-~^ 2 -* 2 > ^ 



^ = H(l-^){(3 + ^aV-(H-^)r 3 } 



+ ^|^(l + ^)r(Z 2 -3^) (36), 



p =s !!4 ( 8 + ^)(fl-_0 + ^ily^(« , -80 (37), 

$? = ^ + ^(l-^)r 2 , (38). 



1891.] Mr Chree, On thin rotating isotropic disks. 209 

For both the annular and the complete disks as and rz are by (22) 
everywhere zero. The expressions for the strains and stresses in a 
complete disk are correctly deduced from those in an annular 
disk by leaving out all terms containing a! 2 . Allowing for the 
change of notation, the solution for the displacements in a com- 
plete disk differs from my previous one 1 only by terms in rff in 
u, z¥ in w and I 2 in 8. Thus it only adds to the strains given by 
the previous solution certain constant terms of order I 2 , and in no 
respect modifies the conclusions derivable from that solution as 
to the mode in which the strains and stresses alter with the 
variables r, z. 

The expressions (32) and (33) for the stresses in an annular 
disk when terms in I 2 and z 2 are neglected agree with those 
which Professor Ewing 2 quotes as obtained by Grossmann 3 . 
They likewise agree with those found by Clerk Maxwell 4 when 
the error in the sign of his equation (59) pointed out by 
Mr J. T. Nicolson 5 is corrected. The expression (31) for the 
radial displacement when terms of order I 2 are neglected is iden- 
tical with that given implicitly or explicitly by Maxwell, and by 
Grossmann putting his N z = 0, and to the same degree of approxi- 
mation (30) coincides with the value for the longitudinal displace- 
ment to which Maxwell's theory would lead if fully worked. out. 
I shall thus for brevity speak of the expressions our solution 
supplies both for the complete and annular disks when terms of 
order I 2 are neglected as constituting the Maxwell solution. 

The conclusion we are led to is that the methods of Maxwell 
and Grossmann — which seem practically identical — while involving 
inconsistencies 6 and certainly inconclusive from a strict theoretical 
standpoint, perhaps even "paradoxical" as Professor Pearson 7 
states, yet lead to results which if the present investigation can 
be trusted are sufficiently exact for practical purposes so long as 
the disk is very thin. 

From (33) it is obvious that w> is everywhere greater than 5r 
in an annular disk. The same result follows from (38) for a com- 
plete disk, except at the axis where the two stresses are equal. 

1 Quarterly Journal of Pure and Applied Mathematics, Vol. xxm. 1889, Equa- 
tions (129), p. 28. 

2 Nature, 1891, p. 462. 

3 Verhandhmgen des Vereins zur Beforderung des Gcwerbfleisses, Berlin, 1883 
pp. 216—226. 

4 Transactions of the Royal Society of Edinburgh, Vol. xx. Part i., 1853, pp. 
Ill — 112; or Scientific Papers, Vol. i. p. 61. For corrections to Maxwell's second 
equation (57) see Todhunter and Pearson's History of Elasticity, Vol. i. ft. -note 
p. 827. 

5 Nature, 1891, p. 514. 

6 They lead to results inconsistent with one or both of the original assumptions, 
viz. that rz is everywhere zero, and that zz if not also zero is independent of r and z 

7 Nature, 1891, p. 488. 



210 Mr Chree, On thin rotating isotropic disks. [May 4, 

Also zz and rz vanish at every point, thus both in the annular 
and in the complete disk u is everywhere the stress-difference 
and u/r the greatest strain. Both quantities for any given value 
of r are greatest when z = 0, and for any given value of z are 
greatest when r = for the complete disk, or r = a for the 
annular. They are thus according to the solution greatest in 
the central plane, at the centre of a complete disk and at the 
inner edge of an annular. 

According, however, to the principle of statically equivalent 
surface forces our solution does not strictly apply for values of r 
which differ from a or from a' by quantities which do not exceed 
several multiples of I. In other words, it possibly may give values 
for the strains and stresses over the edges differing from the true 
values by terms of the order f. Thus in determining the greatest 
values of the stress -difference or greatest strain, which occur 
at or immediately adjacent to the inner edge in an annular 
disk, we are not warranted in retaining terms of this order of 
small quantities. I thus propose in determining these quantities 
to neglect terms of this order as being of doubtful accuracy, at 
least in an annular disk, and of insignificant magnitude in any 
thin disk. It should be noticed, however, that our complete 
solution gives at all radial distances larger values for the strains 
and stresses in the central plane than when terms in I 2 are neg- 
lected. Thus it would certainly only be prudent to regard the 
values we are about to find from the Maxwell solution for the 
maximum stress-difference and greatest strain as minima, which 
in all probability are exceeded in any actual case. In a thin 
disk, however, the true values can exceed these only by small 
terms, of order (//a) 2 at least. 

Neglecting then terms in I 2 and z\ we find for the maximum 
stress-difference 8 and the largest value of the greatest strain s — 

In a complete disk, ) * s _ 

F \Es, = (1- V )S 1 (40), 

In an annular disk, 8 2 = E\ = \a> 2 P [a 2 (3 + y) + a 2 (1 - y)} . . .(41). 

Since 8 2 = E\ the maximum stress-difference and greatest strain 
theories lead to identically the same result for the so-called 
"tendency to rupture" — i.e. approach to limit of linear elasticity 
— in the annular disk. In the complete disk the maximum stress- 
difference theory assigns for all possible values of y, except 0, 
a lower limit than the other for the safe velocity of rotation. 

Supposing g) 5 and w 2 the limiting safe angular velocities in a 
complete and in an annular disk of the same material and external 
radius, then putting in succession 8 % = 8 2 and s t =s ? , we find 



18.91.] Mr Chree, On thin rotating isotropic disks. 211 

On the maximum stress-difference theory 

1 — rj a' 



>>'= 2 ( 1+ ^) <42) - 



On the greatest strain theory 

»>„-2- (1 _ v)(s + v) (43). 

When (a' /a) 2 is negligible, or there is only a very small axial hole, 
these give respectively 

&)j/&) 2 = a/2 for all values of in, 



a>Jta z = J2/(1 — rf), or nearly 1*633 for 77 = -25. 

The former result was given by Professor Ewing in Nature, and 
he also directed attention to the great diminution it represents 
in the strength of a disk due to the removal of a small axial 
core. The effect is even more striking on the greatest strain 
theory for ordinary values of rj. 

Since the striking character of this result may arouse doubts 
in some minds as to the validity of any investigation which leads 
to it, I would point out that it is not an isolated fact. The 
removal of a central spherical core of any radius however small 
from a sphere rotating about a diameter has, as I have shown 
in a previous paper 1 , a precisely similar effect, increasing very 
largely the greatest values both of the stress-difference and great- 
est strain. The same result also follows the removal of a thin 
axial core from a rotating right circular cylinder whose length 
is constrained to remain constant 2 . 

In discussing the nature of the strains and stresses we may 
for most purposes leave out of account in the first place terms of 
order I 2 or V, regarding them in the light of small corrections to 
the principal terms. 

According to the Maxwell solution, every originally plane 
section parallel to the faces of a complete or annular disk ap- 
proaches at every point the central section, z = 0, and assumes 
the form of a paraboloid of revolution about the axis of rotation. 
In this respect the phenomena are precisely similar to those 
presented by a flat oblate spheroid rotating about its axis of 
symmetry 3 . 

The latus rectum of the paraboloid into which is transformed 

1 Transaction*, Vol. xiv. pp. 467—83. See Tables IV. and VIII. and their 
discussion. 

2 Transactions, Vol. xiv. p. 339. 

3 Transactions, Vol. xv. pp. 10 — 13. 

VOL. VII. PT. IV. 17 



212 Mr Ghree, On thin rotating isotropic disks. [May 4, 

an originally plane section at distance z from the central plane is 
in both the complete and annular disks 

2E+{>n(i + V )a>*pz}. 

It thus depends merely on the angular velocity, the nature of the 
material and the original distance from the central plane. Its 
reciprocal, and so the curvature at the vertex of the paraboloid, 
varies directly as the square of the angular velocity and as the 
distance from the central plane. 

The amount by which the axial point on an originally plane 
section parallel to the faces — of course an imaginary point in an 
annular disk — approaches the central plane varies as the square 
of the radius of the complete disk. The corresponding approach 
in an annular disk varies as the sum of the squares of the radii 
of its edges, and is greater than in a complete disk of the 
same external radius. - The magnitudes of the reductions in the 
thickness, 21, of the disks will be seen from the following data — 

a,t inner edge ^ (o 2 plr) {(3 + rj) a 2 + (1 — rj) a' 2 }, 
In annular disk -| 

at outer edge ^ a? ply {(3 + rj) a' 2 + (1 — rj) a 2 }, 



In complete disk 



at axis ^=, a> 2 pln (3 + rj) a 2 , 

at outer edge jr-^, o> 2 plr) (1 — rj) a 2 



The terms in zl 2 and z 3 in (30) and (35) cut out when z = ± I, 
so that as regards the preceding results as to the change of thick- 
ness there is an exact agreement between the Maxwell solution 
and the more complete solution. 

It will be noticed that the reduction in thickness at the inner 
edge of an annular disk equals the reduction at the axis of a 
complete disk equal in radius to its outer edge, together with the 
reduction at the rim of a complete disk equal in radius to its 
inner edge, the thicknesses, materials and angular velocities being 
the same in each case. Also the reduction in thickness at the 
outer edge of an annular disk exceeds what it would be if the 
disk were complete by the reduction at the axis of a complete 
disk equal in radius to its inner edge. 

The longitudinal strain dw/dz parallel to the axis is a com- 
pression at every point, unless rj = 0, both in the complete and 
annular disks, and diminishes numerically as r increases. For 
ordinary materials it is a quantity of the same order of magnitude 
as the radial and transverse strains, but it vanishes throughout if 

7? = 0. 



1891.] Mr Chree, On thin rotating isotropic disks. 213 

The terms in zl 2 and z 3 in w would indicate that the longi- 
tudinal compression is somewhat greater near the central plane 
and somewhat less near the faces of the disk than according to 
the Maxwell solution. 

For a first approximation confining our attention to the Max- 
well solution in (31) and (36), we see that every point in the 
disk, whether complete or annular, increases its distance from the 
axis, and the transverse strain u/r is thus everywhere an extension. 

In the complete disk the radial strain is an extension inside 
and a compression outside of a cylindrical surface co-axial with 
the disk, and of radius r v given by 

ri = a^(S+ v )/(l+ v ) (44). 

This gives a value for r x less than a for all possible values of tj 
except 0. Any annulus of the disk increases or diminishes in radial 
thickness according as it lies inside or outside this surface. 

In the annular disk the increases da and da in the radii of 
the edges, and d (a — a') in the radial thickness, a — a', are given 

by 

da = ( ^\(l- v )a 2 + (3+ v )a' 2 } (45), 

da = ^ {(3 + V ) a 2 + (1 - v ) a' 2 } (46), 

d ( Ct - a') = «> 2 P &-<*') {(a _ aJ _ v{a + a y } . . (47) . 

Thus the radial thickness is increased or diminished according as 

a/a< or > (I - ^77) 4- (1 + <f v ) (48). 

For the ratio of the radii in the annular disk whose radial thick- 
ness is unaltered, we find 

r) = 0, a'ja = 1, 

7} = "25, a'ja = 3, 

7) = "36, a'ja = '25, 

7} = '5, a' '/a = 1716 approx. 

The radial strain given by the Maxwell solution in (31) is 

£K>-> m 

where f(r) = (1 - V )(S + v )(a* + a' 2 ) - 3 (1 - v 2 ) r 2 

- (l+7?)(3 + 77) ttVV 2 ... (50). 
Since /(a) = - 2*, {(1 - v ) a 2 + (3 + v ) a"}, 

and /(«') = - 2 V {( 1 - 7 1 ) a' 2 + (3 + rj) d% 

17—2 



214 



Mr Ghree, On thin rotating isotropic disks. [May 4, 



we see that the radial strain is a compression at both edges for 
all ratios of a' : a, and for all possible values of 77 except 0. For 
77 = 0, 

/(r) = ^(r 2 -a> 2 -r 2 ) (51), 

and so the radial strain is for all ratios of a : a an extension 
everywhere except at the edges, where it vanishes. 

For all other values of 77 there are always portions of the 
disk immediately adjacent to the edges wherein the radial strain 
is a compression, and it is easily proved that the radial strain is 
everywhere a compression when 

a' /a > 7(1-^(3 + 77) -=- {3^ (1 + 77) + V477 (2 + 77)} . . .(52). 

An idea of the nature of the radial strain under various conditions 
for the value *25 of 77 may be derived from the following table : 



Table I. 



du 



Sign of -=- , and loci where it vanishes for 77 = '25. 



Value of a' /a 
•1 


r/a = 


du 
dr 




r - + 
•1 130 




•927 1 


•2 




•2 -264 


•912 1 


•3 




•3 -409 


■882 1 


•4 




•4 -597 


•806 1 


>-426 




Everywhere a 


compression. 



From the Maxwell solution in (32) and (37) it is obvious that the 
radial stress is a traction for all possible values of 77 at all points 
not on the edge or edges of the disk. 

The terms in rl 2 and rz* in (31)— (33) and (36)— (38) show 
that the complete solution gives algebraically greater values for 
the radial and transverse displacements, strains and stresses at 
all axial distances in the central plane than the Maxwell solution. 
It will be noticed, however, that the mean of each of these 



1891.] Mr Chree, On thin rotating isotropic disks. 215 

quantities taken between the limits — I and +1 of z is the same 
as when terms of order l l are neglected. Thus the Maxwell- 
Grossmann method of solution leads to values for all the radial 
and transverse strains and stresses which are identical with those 
which the present solution supplies for mean values taken through- 
out the thickness of the disk. It also as we have seen when 
fully worked out supplies the same value for w at the plane 
surfaces of the disk, and so the same mean value for the longitu- 
dinal compression throughout the thickness. 

In consequence of the second relation (22) the mutual inclina- 
tions of all material lines in the disk remain unchanged. Thus 
what were originally cylindrical surfaces co-axial with the original 
edges cut orthogonally the surfaces into which have been trans- 
formed what were originally planes parallel to the faces ; i. e. they 
become orthogonal to what are practically a series of paraboloids, 
whose common axis is that about which the rotation takes place. 
This perhaps will convey the clearest idea of how material lines 
originally perpendicular to the faces become under rotation con- 
cave to the axis of the disk. 



May 18, 1891. 

Professor Liveing, Vice-President, in the Chair. 

The following Communications were made to the Society: 

(1) On Parasitic Mollusca. By A. H. Cooke, M.A., King's 
College. 

[Received July 18, 1891.] 

Various grades of parasitism occur among the Mollusca, from 
the true parasite, living and nourishing itself on the tissues and 
secretions of its host, to simple cases of commensalism. Some 
authors have divided these forms into endo- and ecto-parasites, 
according as they live inside or outside of their host. Such a 
division, however, is hardly tenable. Certain forms are indif- 
ferently endo- and ecto-parasitical, while others are ecto-parasitic 
in the young form, and become endo-parasitic in the adult. It 
will be convenient therefore, simply to group the different forms 
according to the home on which they find a lodgement. 

On Ccelenterata. (a) Sponges. Vulsella and Grenatula almost 
invariably occur in large masses of irregular shape, boring into 
sponges, (b) Corals. These form a favourite home of many 
species, amongst which are several forms of Coralliophila, Rhizo- 
chilus, Leptoconcha, and Sistrum. The common Magilus, from the 
Red Sea and Indian Ocean, in the young form is shaped like a 



216 Mr Cooke, On Parasitic Mollusca. [May 18, 

small Buccinum. As the coral (Meandrina) to which it attaches 
itself grows, it develops at the mouth a long calcareous tube, the 
aperture of which keeps pace with the growth of the coral, 
and prevents the mollusc from being entombed. The animal 
lives at the free end of the tube, and is thus continually shifting 
its position, while the space it abandons becomes completely 
closed by calcareous matter. Certain species of Ovula inhabit 
Gorgoniae, assuming the colour, yellow or red, of their host, and, 
in certain cases, developing for prehensile purposes a pointed 
extension of the two extremities of the shell. Pedicularia in- 
habits the common Melithaea rubra of the Mediterranean, and 
another species has been noticed by Graeffe 1 on M. ochracea in 

Fi J l 

On Echinodermata. (a) Grinoidea. Stilina comatuhcola lives 

on Comatula mediterranea, fixed to the outer skin, which it pene- 
trates by a very long proboscis ; the shell is quite transparent 2 . 
A curious case of a fossil parasite has been noticed by Roberts 3 . 
A Calytraea-sh&iped shell named Platyceras always occurred on 
the ventral side of a crinoid, encompassed by the arms. For 
some time this was thought to afford conclusive proof of the 
rapacity and carnivorous habits of the echinoderm, which had 
died in the act of seizing its prey. Subsequent investigations, 
however, showed that in all the cases noticed (about 150) the 
Platyceras covered the anal opening of the crinoid in such a way 
that the mouth of the mollusc must have been directly over the 
orifice of the anus, (b) Asteroidea. The comparatively soft tex- 
ture of the skin of the starfishes renders them a favourite home 
of various parasites. The brothers Sarasin noticed 4 a species of 
Stilifer encysted on the rays of Linckia multiformis. Each shell 
was enveloped up to the apex, which just projected from a 
hole at the top of the cyst. The proboscis was long, and at its 
base was a kind of false mantle, which appeared to possess a 
pumping action. On the under side of the rays of the same 
starfish occurred a capuliform mollusc (Thyca ectoconcha), fur- 
nished with a muscular plate, whose cuticular surface was indented 
in such a way as to grip the skin of the Linckia. This plate 
was furnished with a hole, through which the pharynx pro- 
jected into the texture of the starfish, acting as a proboscis and 
apparently furnished with a kind of pumping or sucking action. 
Adams and Reeve 5 describe Pileopsis astericola as living ' on the 
tubercle of a starfish,' and Stilifer astericola, from the coast of 

1 Described as a Cypraea, but no doubt an Ovula or Pedicularia: C. B. Bakt. 
Par. v. 543. 

2 Von Graff, Z. Wiss. Zool. xxv. 124. 

3 Proc. Avier. Phil. Soc. xxv. 231. 

4 Ergeb. naturw. Forsch. Ceylon, abstr. in J. R. 31. S. (2) vi. 412. 

5 Voyage of the Samarang, Moll. p. 69, PI. xi. f. 1 ; p. 47, PI. xvn. f. 5. 



1891.] Mr Cooke, On Parasitic Mollusca, 2~\ 7" 

Borneo, as ' living in the body of a starfish.' In the British 
Museum there is a specimen of Pileopsis crystallina in situ on 
the ray of a starfish. On the brittle starfishes {Ophiuridae) occur 
several species of Stiliferina. (c) Echinoidea. Various species of 
Stilifer occur on the ventral spines of echinoderms, and are some- 
times found imbedded in the spines themselves. St. Turtoni 
occurs on the British coasts on several species of Echini, and 
Montacuta substriata frequents Spatangus purpureus and certain 
species of Amphidetus, Cidaris and Brissus. Lepton parasiticum 
has been described from Kerguelen I. on a Hemiaster, and a new 
genus, Robillardia, has recently been established 1 for a Hyalinia- 
shaped shell, parasitic on an Echinus from Mauritius, (d) Holo- 
thuroidea. The 'sea-cucumbers' afford lodgement to a variety 
of curious forms, some of which have experienced such modi- 
fications that their generic position is by no means established. 
Entoconcha occurs fixed by its buccal end to the blood-vessels of 
certain Synaptae in the Mediterranean and the Philippines. Ento- 
colax has been dredged from 180 fath. in Behring's Straits, attached 
by its head to certain anterior muscles of a Myriotrochus 2 . A 
curious case of parasitism is described by Voeltzkow 3 as occurring 
on a Synapta found between tide-marks on the I. of Zanzibar. 
In the oesophagus of the Synapta was found a small bivalve 
(Entovalva), the animal of which was very large for its shell, and 
almost entirely enveloped the valves by the mantle. As many as 
five specimens occurred on a single Synapta. In the gut of the 
same holothurian lived a small univalve, not creeping freely, but 
fixed to a portion of the stomach wall by a very long proboscis 
which pierced through it into the body cavity. This proboscis 
was nearly three times as long as the animal, and the forward 
portion of it was set with sharp thorns, no doubt to make it to 
retain its hold and resist evacuation. Various species of Eulima 
have been noticed in every part of the world, from Norway to 
the Philippines, both inside and outside Holothurians 4 . Stilifer 
also occurs on this section of echinoderms 5 . 

On Annelida. Cocliliolepas parasiticus has been noticed under 
the scales of A coetes lupina (a kind of ' sea-mouse') in Charleston 
Harbour 6 . 

On Crustacea. A mussel, § in. long, has been found 7 living 
under the carapace of the common shore-crab (Carcinus maenas), 
but this is not so much a case of parasitism as of involuntary 

1 E. A. Smith, Ann. Mag. Nat. Hist. 1889 (i), 270. 

2 Journ. de Conch. (3) xxrx. 101. 

3 Zool. Jahrb. Abth. f. Syst. v. 619. 

4 See especially Semper, Animal Life, Ed. 1, p. 351. 

5 Gould, Moll, of U.S. expl. exped. 1852, p. 207 (St. acicula, from Fiji). 
« Stimpson, Proc. Bost. Soc. iV. H. vi., 1858, p. 308. 

7 Pidgeon, Nature, xxxix. p. 127. 



218 Mr Cooke, On Parasitic Mollusca. [May 18, 

habitat, the mussel no doubt having become involved in the 
branchiae of the crab in the larval form. 

On Mollusca. A species of Odostomia (pallida Mont.) is found 
on our own coasts on the 'ears' of Pecten maximus, and also 1 on the 
operculum of Turritella communis. At Panama the present writer 
found Crepidula (2 sp.) plentiful on the opercula of the great 
Strombus galea and of Cerithium irroratum. Amalthea is very 
commonly found in Conus, Turbo, and other large-sized shells, but 
this is probably not a case of parasitism, but simply of con- 
venience of habitat, just as young oysters are frequently seen on 
the carapace and even on the legs of large crabs. 

On Tunicata. Lamellaria is said to deposit its eggs on an 
Ascidian (Leptoclinum), and the common Modiolaria marmorata 
lives in colonies imbedded in the tegument of Ascidia mentula 
and other simple Ascidians. 

Special points of interest with regard to parasitic mollusca 
relate to (1) Colour. This is in most cases absent, the shell being 
of a uniform hyaline or milky white. This may be due, in the 
case of the endo-parasitic forms, to absence of light, and possibly, 
in those living outside their host, to some deficiency in the 
nutritive material. A colourless shell is not necessarily pro- 
tective, for though a transparent shell might evade detection, 
a milk-white hue would probably be conspicuous. (2) Modifica- 
tions of structure. These are in many cases considerable. Ento- 
concha and Entocolax have no respiratory or circulatory organs 
and no nervous system ; Thyca and certain Stiliferi possess a 
curious suctorial apparatus ; the foot in many cases has aborted, 
since the necessity for locomotion is reduced to a minimum 2 , and 
its place is supplied by an enormous development of the proboscis, 
which enables the creature to provide itself with nutriment with- 
out shifting its position. Special provision for holding on is 
noticed in certain cases, reminding us of similar provision in 
human parasites. Eyes are frequently, but not always wanting, 
even in endo-parasitic forms. A specially interesting modification 
of structure occurs in ,(3) the Radida. In most cases (Eidima, 
Stilifer, Odostomia, Entoconcha, Entocolax, Magilus, Coralliophila, 
Leptoconcha) it is absent altogether. In Ovula and Pedicidaria, 
genera which are in all other respects closely allied to Cypraea, 
the radula exhibits marked differences from the typical radula of 
the Cypraeidae. The formula (3'1\3) remains the same, but the 
laterals are greatly produced and become fimbriated, sometimes 
at the extremity only, sometimes along the whole length. A 

1 Smart, Journal of Conch, v. 152. 

2 Semper notices a case where a Eulima whose habitat is the stomach of a Holo- 
thurian retains the foot unmodified, while a species occurring on the outer skin, but 
provided with a long proboscis, has lost its foot altogether. 



1891.] Mr Cooke, On Parasitic Mollusoa. 219 

very similar modification occurs in the radula of Sistrum spectrum 
Reeve, a species which is known to live parasitically on one of 
the branching corals. Here the laterals differ from those of the 
typical Purpuridae in being very long and curved at the ex- 
tremity. The general effect of these modifications appears to 
be the production of a radula rather of the type of the vegetable- 
feeding Trochidae, which may perhaps be regarded as a link in 
the chain of gradually degraded forms which eventually terminate 
in the absence of the organ altogether. The softer the food, the 
less necessity there is for strong teeth to tear it ; the teeth either 
become smaller and more numerous, or else longer and more 
slender, and eventually pass away altogether. It is curious, how- 
ever, that the same modified form of radula should appear in 
species of Ovula (e.g. ovum), and that the same absence of radula 
should occur in species of Eidima (e.g. polita) known not to be 
parasitical. This fact perhaps points back to a time when the 
ancestral forms of each group were parasitical and whose radulae 
were modified or wanting, the modification or absence of that 
organ being continued in some of their non-parasitical descendants. 

(2) Exhibition of models of double supernumerary appendages 
in Insects : also of a mechanical method of demonstrating the 
system upon which the Symmetry of such appendages is usually 
arranged. By W. Bateson, M.A., St John's College. 

(3) On the nature of the excretory processes in Marine Polyzoa. 
By S. F. Harmer, M.A., King's College. 

[Abstract: reprinted from the Cambridge University Reporter, May 26, 1891.] 

This communication was the result of an occupation of a 
University table at the Zoological Station at Naples during the 
Easter Vacation of 1891. 

Observations were made on the manner in which various 
artificial pigments were excreted in Bugula and in Flustra, on 
the lines adopted by Kowalevsky (Biolog. Gentralblatt, ix., 1889 — 
1890, pp. 33 etc.) for other Invertebrates. The general result of 
the experiments was to show that excretion is not performed by 
organs comparable with nephridia, but that this process is carried 
on by free mesoderm cells, and to some extent by the connective 
tissue and by the walls of the alimentary canal. Evidence was 
obtained to show that the periodic loss of the alimentary canals 
leading to the formation of the " brown bodies " may be regarded 
as, to some extent, an excretory process. 



220 Mr Brown, On the part of the 'parallactic [June 1, 

June 1, 1891. 
Prof. G. H. Darwin, President, in the Chair. 
The following communications were made to the Society : 

(1) On the part of the 'parallactic class of inequalities in the 
moon's motion, which is a function of the ratio of the mean motions 
of the sun and moon. By Ernest W. Brown, M.A., Fellow of 
Christ's College. 

In a paper to be published shortly, a solution will be given by 
approximation in series, of the equations for this class of in- 
equalities. In Vol. I. of the American Journal of Mathematics, 
Mr G. W. Hill has shown that by using rectangular instead of 
polar co-ordinates, the inequalities depending only on the mean 
motions of the sun and moon can be found to a high degree of 
accuracy with comparatively little trouble ; and that the series to 
be obtained may be rendered more convergent by developing in 
terms of n'/(n — n) — m', instead of n'/n = m as has been done 
in most of the previous theories. He further shows that by 
developing in terms of m'/(l — ^m) = //. a still greater degree of 
convergency is obtained. The results are expressed in rectangular 
co-ordinates and on that account are not convenient for obtaining 
the algebraical expressions of the longitude and parallax of the 
moon. But these transformations will still have force when we 
change to polar co-ordinates, so that by putting in Delaunay's 
series for this class of inequalities m= m'/(l +m') = //-/(l +!/"■) 
and expanding in powers of m or /u, we should get a better ap- 
proximation to the truth. This is not necessary in the Variation 
Inequality which Delaunay has found with sufficient accuracy for 
practical purposes. But in the Parallactic Inequality he stops at 
(a/a').m 7 , the numerical multiplier of which is roughly 55113; 
the term expressed in seconds is 0"'38. By substituting 

m = m'/(l + mf) 

and developing in terms of m', this multiplier is reduced to about 
one half its former value ; but since m' is nearly one-twelfth 
greater than m the accuracy is not increased, though the new 
series has greater convergency. 

On using Hill's method with rectangular co-ordinates for the 
parallactic class of inequalities (i.e. those dependent on the ratio 
of the mean distances of the moon and sun) a factor whose value 
is 1/(1 — 4<m' — ...) appears, and to the expansion of this factor in 
powers of m is due the slow convergence of the series giving the 



1891.] clans of inequalities in the moon's motion. 221 

coefficient of the parallactic inequality. Delaunay's series for this 
inequality is 1 

- % [-^m + $£m 2 + ^-m 3 + 1 % 9 - 7 -m 4 + ^Hf-^m 5 + 63 |g|| 13 m 6 

74"'02 34f"-33 11"'89 4"-43 l"-86 0"-7l 

_L 10835 53 7 1 59 ,»/] 
~ 1116608 "* J' 

0"*38 

where the coefficients expressed in seconds of arc are written 
below. In this expression put m = m'/(l + in) and expand in 
powers of m, subtract from the result the expansion of 

(igsm! + |m /2 )/(l - 4m<) 

(which are the first two terms found by rectangular co-ordinates) 
in powers of in and we obtain instead of Delaunay's series the 
expression 

- % [(¥•»' + f O/a - W) - ftt^ - W#™' 4 + HtfF™' 5 

-118"25 -ll"-47 +l"-83 + 0"-37 -0"-16 

_ 3 1406 5 7 ^/6 , 3320 080 247^/71 
245 7 6 " v ^ T9 66W '"' J" 

+ 0"-02 - 0"-20 

Newcomb' 2 suggests that the last term of Delaunay's expression 
is wrong and this expression which is deduced from Delaunay's 
would seem to support that view. I hope however to verify the 
term. Leaving this term out of consideration the increased con- 
vergency of the series is manifest. The first three terms give 
nearly the whole value of the coefficient. In the paper the other 
coefficients of the periodic inequalities of this class will be dealt 
with in a similar manner, and expansions will be given for the 
whole class of inequalities, the factor 1/(1 —4m' — ...) being intro- 
duced also into the higher powers of in. It will then not be 
necessary to go further than m' 5 a/a' to get the values of the 
coefficients correct to one hundredth of a second of arc; and, for 
this degree of accuracy, by the methods given the approximations, 
either for algebraical or numerical results, are not long. 

(2) On Pascal's Hexagram. By H. W. Richmond. 

The author applies Cremona's method of deriving the hexa- 
gram by projection of the lines on a nodal cubic surface from the 
node. By use of a new form of the equation to this surface the 
equations of the lines are obtained in a perfectly symmetrical 
form, and their properties thence developed. 

1 Memoires de V Academic des Sciences, Tome xxix. p. 847. 

2 Astronomical Papers for use of American Ephemiris, Vol. i. pt. n. p. 71. 



222 



Mr Chree, On some experiments on 



[June 1, 



(3) A Linkage for describing Lemniscates and other Inverses 
of Conic Sections. By R. S. Cole. 

(4) Some experiments on liquid electrodes in vacuum tubes. 
By C. Chree, M.A., Fellow of King's College. 

The experiments discussed in this paper were undertaken 
at the suggestion of Professor J. J. Thomson, in order to throw 
further light on the nature of the electric discharge in a vacuum 
tube with liquid electrodes. To render the work intelligible it 
is necessary to give a brief sketch of the general nature of the 
discharge at low gaseous pressures, and to mention certain results 
of previous observers and certain of their theoretical conclusions. 

At low pressures the phenomena at the cathode — or electrode 
to which the positive current travels in the tube — are the most 
striking, so that it is the most convenient point of departure. 
The phenomena ordinarily observed between the electrodes when 
not too near together are as follows : — 

(A) a thin luminous envelope covering the cathode, 

(B) a much thicker, but still except at very low pressures, 
short dark space, 

(C) a bright, usually blue, space of considerable length, 

(D) a second dark space, 

(E) a more or less luminous interval extending to the anode. 

There is unfortunately no universally accepted English termi- 
nology for these spaces. The following table shows some of the 
terms most commonly used in English, and also the ordinary 
German terminology. 

Table I. 





German 


Schuster 


Spottiswoode 
and moulton 


Ordinary 

USAGE 


(A) 


Lichtsaum 


Narrow layer 






(B) 


dunkle Kathodenraum 


Dark space 


Crookes' space 


Crookes' space 


(C) 


Glimmlicht 

Ihelle Kathodenschicht 
\ Glimmlicht strahlen 


Glow proper 


{ Negative glow 
( Negative haze 




(D) 


dunkle Trennungsraum 


Dark interval 


Negative dark 
space 


Faraday space 


(E) 


positive Licht 


Positive light 


Positive light 


• 



1891.] liquid electrodes in vacuum tubes. 223 

(A), (B) and (C) are regarded as forming the negative light 
or discharge, and (D) as separating the negative and positive 
discharges. (A) is very inconspicuous, if actually existent, at 
gaseous pressures exceeding 1 or 2 mm. of mercury, but at very 
low pressures it is fairly bright though very thin. 

(B) is also insignificant so long as the pressure exceeds a few 
mm. of mercury, but at very low exhaustions it has been ob- 
served to exceed a length of 2 cm. Its length has been shown 
not to depend much on the material of the cathode when metallic. 
It is also usually but little dependent on the strength of the 
current. It varies to a considerable extent with the nature 
of the gas, being according to Professor Crookes 1 decidedly longer 
in hydrogen and shorter in carbonic acid gas than in air. In 
any one gas it is supposed to increase in length as the pressure 
is reduced, so that its magnitude gives a useful if not very exact 
indication of the degree of exhaustion 2 . The following table 
gives some of the measurements of Crookes 3 — altered to mm. — 
and Puluj 4 for air vacua. 













Table II. 








Crookes. 












Puluj. 


Pressure in 


mm. 


Length of 


(B) 


in 


Pressure in mm. Length of (B) in 


ofHg. 






mm. 








ofHg. 


mm. 


•313 






6-5 








1-46 


25 


123 






8-5 








•66 


4-5 


•078 






12-0 








•30 


7-8 


•042 






150 








•24 


9-5 


•020 






25 








•16 
■06 


14-0 
22-0 



For a given length of (B) the pressures found by Crookes 
are very considerably less than those found by Puluj, a dis- 
crepancy ascribable perhaps to differences between their tubes 
and cathodes but due probably in greater measure to the un- 
certainties attending the determination of such low pressures. 

The conditions of Puluj's experiments resembled more closely 
those of the present paper than did Crookes', so the former's 
results seem a priori the best for comparison with those de- 
scribed here. It must, however, be remembered that at the 
lowest pressures in my experiments, the gas in the tube was 
doubtless in great measure, if not almost exclusively, vapour 
from the liquid electrodes. Though not absolutely black, (B) in 

1 Phil. Trans. 1879, pp. 138—9. 

2 Phil. Trans. 1879, p. 137. 

3 Phil. Trans. 1879, pp. 158—9. 

4 Sitzungsberichte Math. Nat. Classe der k. Akad., Bd. lxxxi., Abth. n. Wien, 
1880, p. 874. 



224 Mr Ghree, On some experiments on [June 1, 

general seems so to the eye, and the surface separating it from (C) 
is usually sharply defined. 

(C) has its brightest side next the cathode, and it is by some 
writers divided into a brighter portion, the negative glow, and 
a less bright portion, the negative haze. (C) appears almost as 
soon as the pressure is sufficiently reduced to allow the dis- 
charge to pass. It increases in length 1 as the pressure is re- 
duced within at least certain limits. The transition from (C) 
to (D) is usually so gradual no exact line of separation can be 
drawn. (D) is not always visible. Its presence depends to a 
considerable extent on the strength of the discharge. At low 
pressures an increase in the strength of discharge tends to make 

(D) contract 2 . At high pressures when the discharge first passes 

(E) consists of a succession of twig-like independent discharges, 
which, as the pressure is lowered, transform into what seems to 
the eye a tolerably uniform column whose colour in most gases, 
especially in air, is bright red. As the exhaustion proceeds the 
colour becomes less bright, and the apparent uniformity of the 
column tends to disappear. At moderate pressures such as 1 or 
2 mm. of mercury with a regular source of current, (E) consists 
in general of a succession of similar units, striae, each having a 
sharply defined luminous head on the side next the cathode, and 
gradually fading on the side next the anode into a seemingly 
non-luminous portion. At very low pressures the striae are 
sometimes very faintly defined, if existent, and (E) appears as 
a hazy luminosity, generally of a blue tint. 

Taking the simplest case, viz. a long, uniform, straight cylin- 
drical tube with flat metal electrodes, whose planes are per- 
pendicular to the axis of the tube, and whose diameters are 
not very small, each stria in the positive column has in ana- 
tomical language an opisthocoelous form, the convex surface being 
that of the luminous head. This surface is, however, in general 
of smaller curvature than a sphere of diameter equal to that 
of the tube. Whether the positive column be striated or not, 
the surface separating it from (D) has its convexity on the side 
next the cathode. This surface is usually sharply defined when 
(D) appears at all. 

There is another phenomenon whose relation to the discharge 
is somewhat doubtful, viz. the phosphorescence observed in good 
vacua. The more brilliant phosphorescent phenomena are 
beautifully shown by many well-known experiments of Professor 
Crookes. He believes the " molecular streams " or " radiant 
matter " — German " Kathodenstrahlen" — whose incidence on the 

1 After completely covering a wire cathode it expands as the current increases. 
See Hittorf, Wied. Ann. 20, 1883, p. 746. 

2 Hittorf, 1. c. pp. 736—7. 



1891.] liquid electrodes in vacuum tubes. 225 

glass of the tube sets up this luminosity, to be gas molecules 
charged at the cathode and projected with great velocity at 
right angles to its surface. According to Goldstein and others 
the direction of projection varies to some extent from the 
normal. Professor Crookes 1 appears to have originally thought 
these molecular streams peculiar to very low vacua and indicative 
of a fourth state of matter, bearing to the gaseous state some- 
what the same relation as it bears to the liquid. Messrs Spottis- 
woode and Moulton 2 have, however, shown that phosphorescence 
can be produced at quite high pressures provided the intensity 
of the negative discharge be sufficiently increased. 

These latter observers 3 have treated with great fulness other 
less conspicuous phosphorescent effects. They found that under 
certain conditions a portion of the wall of a vacuum tube touched 
by the finger or having an earthed conductor in its neighbour- 
hood acts as a sort of secondary cathode, setting up phosphor- 
escence on the opposite side of the tube. They also found that 
at very low pressures the positive discharge when in the form 
of a hazy luminous column, occupying in general only a portion 
of the tube's cross-section, creates a sort of demand for negative 
electricity which may be supplied by a discharge proceeding from 
the walls of the tube in directions at right angles to its length 
and creating phosphorescence. Further when the positive column 
is cut at an angle by the wall of the tube, as at a sharp bend, 
phosphorescence appears whose position is as if it were due to 
the impact of molecules travelling along the positive column 
in the direction from cathode to anode. This latter phenomenon 
has been more exhaustively treated by Goldstein 4 . He found 
that at very low pressures in a tube bent at right angles any 
number of times, there is at every bend between the cathode 
and anode where the positive column extends, a patch of phos- 
phorescence, situated as if due to rays travelling from the cathode 
to the anode. Supposing an electrode at A in a tube AD at 
right angles to a tube BDE, and that the latter tube is closed 
at the end B, while a third tube at right angles to it leads 
from E to an electrode G, then according to Goldstein, if I in- 
terpret him correctly, there is phosphorescence at E whether 
the cathode be A or C, but none at B unless C is cathode. 

Goldstein 5 also describes the production of phosphorescence 
at positions which could not be reached by molecular streams 
travelling in straight lines from the cathode, which he apparently 

1 Phil. Trans. 1879, pp. 142, 143, 163, 164, etc. 

2 Phil. Trans. 1880, pp. 582—6, etc. 

3 1. c, pp. 602—6, 616—7, 620, etc. 

4 Wied. Ann. 12, 1881, pp. 104—9, and figs. 16 and 17, Taf. i. 

5 Wied. Ann. 15, 1882, pp. 246—254, 



226 Mr Chree, On some experiments on [June 1, 

ascribes to a species of diffuse reflexion at a bend in a narrow 
part of the tube. Perhaps this is immediately connected with 
another phenomenon he observed \ viz. that if the vacuum tube 
be of variable section then a place where in passing from cathode 
to anode there is a sudden large increase in the diameter takes 
upon itself the functions of a secondary cathode, dominating the 
character of the striae on the side next the anode and emitting 
molecular streams along the axis of the tube. 

The view that phosphorescence is due to the impact of finely 
divided matter torn off the cathode by the current has been 
maintained by Puluj 2 and others, but it seems obviously in- 
applicable to the phosphorescence proceeding from a secondary 
cathode, or to that called in by the positive column. Even as 
concerns the ordinary phenomena some experiments devised by 
Crookes 3 seem very adverse to Puluj's view. It is, however, 
unquestionable that cathodes of most substances have matter 
torn off by the discharge, and that it is frequently largely de- 
posited on those portions of the tube where the ordinary phos- 
phorescence is most conspicuous. Messrs Spottiswoode and 
Moulton 4 have pointed out other resemblances between the action 
of the molecular streams and that of matter such as lamp-black 
actually projected from the cathode. 

As regards the function of the molecular streams in the 
discharge no final results have been obtained, though much 
speculation exists. Messrs Spottiswoode and Moulton give as 
the result of their investigations: — "At present we have come 
to no definite conclusion . . . , but we cannot say that we are 
aware of anything that conclusively shows that they (the mole- 
cular streams) have any definite electrical function to perform 
in the discharge," 1. c. p. 650. " The most attractive hypothesis re- 
lating to their functions is that they officiate at the birth of the 
discharge and enable it to get into the gaseous medium...," 
1. c. p. 651. 

From the remainder of their cautiously worded remarks, I 
believe Messrs Spottiswoode and Moulton would regard the 
function ascribed to the molecular streams by this " attractive 
hypothesis " to be that of carriers of a convective discharge out 
into the gas. A slight modification would however adapt the 
hypothesis to the very suggestive views of Messrs E. Wiedemann 
and H. Ebert 5 . 

1 Wied. Ann. 11, 1880, p. 836, and Ann. 12, 1881, p. 276. 

2 Wiener, Sitzungsberichte, Bd. lxxxi., Abth. n., 1880, pp. 864 — 923, see specially 
p. 873. 

3 Journal of Electrical Engineers, Vol. xx., Feb. 1891, pp. 29 — 36. 

4 Phil. Trans. 1880, pp. 582 and 649—650. 

5 Wied. Ann. 35, 1888, pp. 217 and 258—9. 



1891.] liquid electrodes in vacuum tubes. 227 

These observers found that the illumination of the cathode 
surface by ultra-violet light, which in general has at high gaseous 
pressures a remarkable effect in facilitating the discharge, ceases 
to have any certain effect at low pressures when the presence 
of phosphorescence shows the existence of molecular streams. 

Their explanation is that ultra-violet light promotes the dis- 
charge by setting up at the cathode surface, apparently in the 
ether, vibrations of great rapidity such as would be produced 
by heating the cathode to a high temperature — in general a 
most efficacious way of promoting discharge — , but that the mole- 
cular streams are the consequence or the concomitant of the 
production of such rapid vibrations by some independent cause 
which they do not specify. Thus the function which the ultra- 
violet light performs at high pressures, is already fully provided 
for at low pressures. 

The point I more specially wish to bring out is this. At 
high pressures whatever tends to increase the violence of the 
negative discharge — e.g. an air spark in the circuit on the cathode 
side of the tube — tends to set up the molecular streams. There 
seems, however, strong grounds for believing that the production 
of these streams actually facilitates the discharge, rendering it 
less violent and disruptive than it otherwise would be, especially 
at very low pressures. 

As regards the nature of the discharge itself various views 
are entertained, to some of which reference is necessary to explain 
what follows. In the opinion of Professors J. J. Thomson 1 , 
Schuster 2 and others the ordinary vacuum tube discharge is 
in a way electrolytic. The molecules of gas become dissociated 
and the atoms act as carriers of positive and negative electricity. 
The heating of the cathode or anode, as the case may be, or the 
presence of a heated wire, influences which have been shown by 
Hittorf 3 , Elster and Geitel 4 and others to promote discharge 
in a wonderful way, operate on this theory by dissociating the 
gas and so furnishing atoms ready to respond at once to the 
directive action of the external source of electricity. At a 
luminous part of the discharge there is a production of heat 
owing to the coming together of oppositely charged atoms, at a 
dark part such coalitions are rare. Other writers, e.g. Lehmann 5 
and E. Wiedemann 6 , regard the discharge as usually of two co- 

1 Phil. Mag. Vol. xv., 1883, pp. 427—434; Aug. 1890, pp. 129—140; March 
1891, pp. 149—171, etc. 

2 Proceedings of the Roijal Society, Vol. xxxvir., 1884, pp. 317 — 339, and Vol. 
XLifc, 1887, pp. 371—9. 

3 Wied. Ann. 21, 1884, pp. 106—139. 

4 Wied. Ann. 37, 1889, pp. 315—329, and Ann. 38, 1889, pp. 27—39, etc. 

5 Molekularphysik, Bd. n. pp. 220 et seq. 

6 Wied. Ann. 35, 1888, p. 256. 

VOL. VII. PT. IV. 18 



228 Mr Ghree, On some experiments on [June 1, 

existent but more or less independent parts, a dark probably 
convective discharge independent of chemical action, and a lu- 
minous discharge. Lehmann's idea is something of the following 
character. The luminous discharge is essentially disruptive and 
intermittent whatever be the nature of the source of electricity. 
Hittorf 1 and Home'n 2 it is true, employing for the source a large 
number of cells, have imagined they got a steady luminous dis- 
charge through the tube like the current in a metallic conductor. 
But their reasons for this view such as the silence of a telephone 
in the circuit are, according to Lehmann 3 , not conclusive, because 
the steadiness of the current outside the tube does not necessarily 
prevent its being intermittent inside. He regards the electrodes 
as charging and discharging like condensers, requiring a certain 
potential depending on the density, temperature, etc. of the gas 
before the luminous discharge is possible. Simultaneously, how- 
ever, the electrodes leak into the tube, the discharge being 
carried off probably convectively without any necessary lumi- 
nosity. An increased brilliancy in the tube only implies an 
increased quantity of electricity passing at each individual lu- 
minous discharge, and thus it accompanies whatever raises the 
capacity of the electrodes or affords an obstacle to rapid dis- 
charge. This explains the action of an air spark in the circuit 
outside the tube. A diminished brilliancy may mean the passage 
of a smaller current, or it may indicate an increased rapidity in 
the succession of luminous discharges without any alteration in 
the current outside the tube, or it may indicate the operation 
of some agency facilitating the convective discharge. It is to 
one or both of the latter causes that Lehmann would ascribe 
the effects of heating a cathode or exposing it to ultra-violet 
light. It is unquestionable that in many such cases the diminu- 
tion in the luminosity of the tube is very striking. Lehmann 4 
does not regard the luminous discharges at the anode and cathode 
as having any necessary relation in the rapidity of their suc- 
cession. He regards a red colour as merely indicating a strong, 
a blue colour a weak discharge. Thus if, as usual, the positive 
discharge is red and the negative blue, the difference is to be 
accounted for either by the generally smaller cross-section of the 
positive column, or by the interval between successive discharges 
being longer at the anode than at the cathode. 

A good many writers while recognizing a convective discharge 
ascribe it to the action of dust particles 5 . These may exist in 

1 Wied. Ann. 20, 1883, pp. 705—712, etc. 

2 Wied. Ann. 38, 1889, pp. 172 et seq. 

3 Molekularpliysik, Bd. n. pp. 234 — 7. 

4 Molekularpliysik, Bd. n. p. 257. 

5 See Lenard and Wolf, Wied. Ann, 37, 1889, pp. 443—456. 



1891.] liquid electrodes in vacuum tubes. 229 

the original gas or be derived from the cathode by the dis- 
integration which has been shown to accompany the passage of 
the discharge, and in many cases at least to follow the incidence of 
ultra-violet light. 

Whatever the nature of the discharge may be it certainly does 
not follow Ohm's law. Hittorf and Homen, using a current which 
outside the vacuum tube seemed steady and continuous, found 
under certain limitations — depending on the spread of the ne- 
gative glow over the cathode — that the potential difference at 
low pressures between the electrodes was nearly independent of 
the strength of the current. At very low pressures the fall of 
potential took place in great measure quite close to the cathode 
surface. Hittorf 1 found in the positive part of the discharge a 
more or less regular fall of potential, and this fall per unit length 
of tube was much greater than that in the non-luminous Faraday 
space. Thus here at least non-luminosity seems to indicate 
diminished resistance. 

As one of the liquids I tried was mercury some points con- 
nected with discharge through Hg. vapour claim special notice. 
Mercury vapour being usually considered mon-atomic, it is clear 
that discharge through it presents some peculiarities on the 
electrolytic theory of discharge, there being no very obvious 
means of separating the gas into carriers of positive and negative 
electricity. 

This peculiarity drew to it the attention of Professor 
Schuster 2 who states as the result of very careful experiments: 
— "I find that if the mercury vapour is sufficiently free from 
air, the discharge through it shows no negative glow, no dark 
spaces, and no stratifications." He also found the discharge to 
present almost exactly the same features at both electrodes. The 
introduction of a very slight trace of air set up the Crookes' 
dark space at once. Experiments in agreement with Schuster's 
are described by Natterer 3 . Professor Crookes * has, however, 
arrived at results diametrically opposed to those of Professor 
Schuster. After taking the greatest pains to prevent the pre- 
sence of any foreign gas, he found a distinct Crookes' space and 
at least traces of a dark Faraday space. He employed aluminium 
electrodes, while Professor Schuster preferred them of platinum 
with only a small surface exposed. 

Elster and Geitel 5 examining the effect of the presence of 
a heated wire in a tube on the contained gas, found mercury 

1 Wied. Ann. 20, 1883, pp. 726 et seq. 

2 Proceedings of the Royal Society, Vol. xxxvu., 1884, p. 319. 

3 Wied. Aim. 38, 1889, p. 669. 

4 Journal of Electrical Engineers, Vol. xx., Feb. 1891, pp. 44 — 6. 

5 Wied. Ann. 37, 1889, pp. 319 and 327. 

18—2 



230 Mr Chree, On some experiments on [June 1, 

vapour to possess the rare, if not unique, property of showing 
no electrification. This of course fits in extremely well with 
the electrolytic theory, which these writers seem to favour. 

Gassiot 1 more than thirty years ago made some interesting 
experiments on mercury. His apparatus was so constructed that 
he could have for his electrodes either metal wires or the Hg. 
surface itself. Describing the appearance of the discharge as 
the mercury rose in the tube, he says, " as soon as the mercury 
ascends above the negative wire, a beautiful lambent bluish 
white vapour appears to arise, while a deep red stratum becomes 
visible on the surface of the mercury," p. 4. This red glow 
was only sometimes apparent. In one place Gassiot suggests it 
may be due to impurities in the mercury, in another he considers 
it analogous to the glow on a platinum wire electrode, but he 
seems to have arrived at no final conclusion. A mercury surface, 
he says, when cathode is all covered with a luminous white 
glow, but when anode only the extreme point is luminous. He 
records numerous observations on striae in tubes in which 
mercury was present, but it is not always easy to follow the 
exact conditions of the experiment. He mentions that by re- 
ducing the temperature of a vacuum tube containing mercury to 
— 102° F. or raising it to 600° F. he caused the striae to disappear. 
He also on several occasions got rid of striae at ordinary tem- 
peratures by using for cathode the surface of the mercury itself. 
Thus on p. 7, Phil. Trans. 1858, he says, "...immediately it 
(the mercury) covers the negative wire the stratifications dis- 
appear, and the interior of the globe is filled with bluish light." 
I think one may fairly conclude from his experiments that the 
nature of the electrodes and especially of the cathode exercises 
an important influence on the phenomena observed in the dis- 
charge through mercury vapour. 

In the experiments now to be described I used the secondary 
current from an induction coil, varying the primary current ac- 
cording to circumstances. When the resistance in the vacuum 
tube circuit is sufficiently large only the direct extra current 
passes. The appearance of the discharge shows at once whether 
this is the case. 

The tubes which I employed were constructed by Professor 
Thomson's assistant, Mr Everett, who rendered me valuable assist- 
ance in the course of the experiments. As the first tube had 
only a brief existence it will suffice to describe the second which 
resembled it in all essential points. The diagrammatic sketch, 
fig. 1, p. 231, will explain the general character. The nature 
of the electrodes varied but their position in the tube was fairly 

1 Phil. Trans. 1858, pp. 1—16; and 1859, pp. 137—160, 



1891.] 



liquid electrodes in vacuum tubes. 



231 



constant. For one of the vertical tubes, say AE, the distance 
of D above the electrode was 170 mm. of which the lowest 
100 mm. was of uniform diameter, 13"5 mm. externally. Above 
this the tube narrowed for some 12 mm. and then continued of 




uniform external diameter 8 mm. up to E at 250 mm. above 
the electrode. The horizontal tube DF had an external diameter 
of 6 mm. and a total length of 160 mm. Thus the total distance 
between the electrodes was about 50 cm. These measurements 
were not made with any great exactness and the surfaces of the 
electrodes were not kept at a perfectly constant level. The 
tube CHP led to the mercury pump, GH being vertical, and 
HP parallel to DF. This tube was of about the same diameter 
asDE. 

Supposing the tube AD to have a liquid electrode, this was 
connected with the exterior by a fine wire whose top was about 
1 cm. below the liquid surface, and which passed down air-tight 
through the bottom of the tube. 

The point to which Professor Thomson originally directed my 
attention was the question whether there was any change in- 
troduced in the liquid surface by the discharge which altered 
the subsequent character of the luminous appearances. As my 
results on this point are intelligible only when the form of the 
discharge has been described, their discussion is postponed to a 
later part of the paper. 

There were three sets of experiments on mercury. In the 
first both electrodes were mercury surfaces ; in the second one 
was mercury, and the other an uncovered platinum wire ex- 
tending some 2 cm. up the axis of one vertical tube ; in the 
third set the platinum wire was replaced by a flat horizontal 



232 Mr Ghree, On some experiments on [June 1, 

plate of aluminium. The gas originally in the tubes was always 
air, and a mercury pump was used. 

In all three cases the following phenomena were observed 
in the Hg. tube — i.e. the branch containing a mercury electrode. 
At fairly high pressures when the spark first passed freely the 
positive discharge was bright red, as is usual in air. As the 
pressure diminished the colour became whiter, and there ap- 
peared numerous striae, a distinct Faraday space, and a whiteish 
blue negative glow. Before the striae became conspicuous the 
discharge left an Hg. anode from its extreme summit, but as 
the exhaustion proceeded it left from an increasing area, till 
finally the whole surface became luminous. 

As the pressure was diminished there appeared what seemed 
to be a Crookes' space over an Hg. cathode. It was not so clearly 
defined as that space usually is, but the surface separating it from 
the negative glow was at first tolerably distinct. This surface 
was convex like the Hg. surface itself and had a very similar 
curvature. This Crookes' space was, however, by no means very 
dark, being in general of a distinctly red aspect, and on occasions 
almost fiery in appearance. The introduction of a minute air 
bubble into the tube made the boundary of this space appear 
much more distinct. As the pressure was further reduced the 
Crookes' space increased in length and the top of the Faraday 
space moved up the tube. This went on until the Faraday 
space extended to 80 or 90 mm. from an Hg. cathode. At this 
stage the stratification in the tube containing an Hg. anode 
was fairly distinct, the length of a stria being about 1 cm. and 
its luminous portion being nearly pure white. After this stage 
there appeared a change in the character of the phenomena most 
conveniently dealt with in the separate discussion of the different 
sets of electrodes. 

Electrodes both Hg. surfaces. 

When the stage just referred to was reached in this case, 
the variety in the appearances at different parts of the discharge 
tended to disappear, the whole assuming a more or less uniform 
white colour. The Faraday space seemed on some occasions 
certainly absent, and the Crookes' space became, to say the 
least of it, exceedingly indistinct. The difficulty of coming to 
a decision as to the existence of these spaces was much in- 
creased by the deposition of matter on the walls of the tube 
containing the cathode. The nature of this deposit was the 
same as that described in the next case. All I can say with 
certainty is, that the red-black space — assumed to represent 
Crookes' space — did not continue to expand continually as the 



1891.] liquid electrodes in vacuum tubes. 233 

exhaustion proceeded. It seemed when longest to reach as far as 
1 cm. above the cathode, becoming less distinct as it expanded. 
At the lowest pressures reached, however, while the 4 or 5 mm. of 
the tube immediately above the cathode were perhaps somewhat 
darker than the average, the luminosity had certainly attained a 
maximum within not more than 7 or 8 mm. of the Hg. surface. 
The stage at which the Faraday space became indistinct depended 
to some extent on the nature of the break in the induction coil. 
With a rapid break and quiet spark the luminosity fell off and 
the Faraday space remained longer distinct. 

Electrodes Hg. surface and Pt. wire. 

With electrodes so different in form the variations in the 
phenomena cannot be entirely ascribed to difference of material. 
According to Goldstein ' an alteration in the size of the anode 
does not affect the striae in a stratified discharge, but a dimi- 
nution in the size of the cathode while leaving unaltered the 
length of the individual striae increases the distance of the head 
of the positive discharge from the cathode. E. Wiedemann a 
found that the relative facility of discharge between points and 
between plates changes with the pressure. Thus at pressures 
over 1 mm. he found the discharge to pass between points and 
not between plates at the same distance apart, whereas at lower 
pressures it passed most easily between the plates. Lehmann 3 
lays down some apparently general laws as to the effects of 
making blunter a cathode, but he does not always seem con- 
sistent on this point, and I have some doubts as to how far he 
bases his views on experiment. So many secondary influences 
are at work, such as the size of the tube and the distance of 
the electrodes, especially when comparable with their transverse 
dimensions or with the length of Crookes' space, that one would 
hardly expect a 'priori any simple general law to apply. 

At the lower pressures the platinum wire rapidly became 
red hot and the deposit on the tube around it became very 
thick, so that it was impossible to see anything of the Crookes' 
space. If it did exist it must have been considerably less than 
the Crookes' space in the Hg. tube when that was last seen 
distinctly. The comparison instituted was thus between the 
distances from the cathodes of the further extremity of the 
Faraday space, i.e. the head of the positive discharge. The 
following are some of the data obtained, the simultaneous lengths 
of the Crookes' space at the Hg. cathode being given when 
noted : 

i. 12, 1881, p. 275. 

i. 20, 1883, pp. 795—7. 

22, 1884, pp. 320—1. 



i Wied. Ann. 12, 1881, p. 275. 

- Wied. Ann. 20, 1883, pp. 795—7. 

3 Molekularplnjiik, Bel. n. pp. 277—8, and Wied. Ann. 2 



234 



Mr Chree, On some experiments on 



[June 1, 







Table III. 






Hg. 


Cathode 


Pt. Cathode 

. A . 


t 

Length of 

Crookes' 

space. 




A. ^ 

Distance above cathode 

of head of positive 

discharge. 


Distance above cathode 

of head of positive 

discharge. 






30 


13 






35 


14 


4 




60 


20 


5 




65 


20 


7 




75 


20 



The distances are in mm., and are all measured from the 
upper extremity of the cathode. The observations recorded in 
the same horizontal line were taken in immediate succession, 
the make and break regulator of the induction coil remaining 
untouched. The uniformity of the pressure and of the make 
and break was tested by reversing the current twice, which 
showed whether the appearances at the surface which was first 
cathode had altered durirjg the observations. The experiments 
extended over a considerable interval and the tubes were cleaned 
out and refilled more than once, thus the difference between 
the cathodes shown by the table, which was confirmed by nu- 
merous observations in which accurate measurements were not 
taken may, I think, be fully relied on. The table shows that 
the distance above the cathode of the head of the positive 
column was on an average about thrice as much when the 
cathode was mercury as when it was platinum. The difference 
was greater the lower the pressure. At the highest pressures 
when the Faraday space first became distinct no exact measure- 
ments were recorded, but the difference though then not so 
striking as in the table was still conspicuous. As the exhaustion 
proceeded very slowly and both tubes were at intervals heated 
up by a burner, the difference can hardly be attributed to any 
great extent to a difference in the gaseous contents of the two 
vertical tubes. 

The difference may be due to the difference in the material 
of the electrodes or to the dissimilarity of their sizes and shapes. 
According to Goldstein the effect of the difference in size of the 
cathodes should have tended in the opposite direction. For 
reasons explained in treating of the next case, I am inclined to 
attribute the difference in considerable measure to the difference 
in shape. 



1891.] liquid electrodes in vacuum tubes. 235 

Appearance of Discharge. 

The tube containing the Pt. electrode always showed a redder 
tint than that containing the Hg. electrode under similar con- 
ditions. 

At a certain stage of exhaustion the former tube was some- 
times the redder even when the Hg. surface was the anode. 
At the lowest pressures, however, the difference between the 
colour of the tubes was inconspicuous, both being prevailingly 
white. At the highest pressures the spark left the tip of the 
Pt. anode and the extreme summit of the Hg. anode, but as 
the exhaustion proceeded it gradually extended down the Pt. 
anode and spread over the surface of the Hg. anode. Eventually 
the positive column covered the whole of the Hg. surface, but 
within 1 or 2 mm. up the tube it had contracted somewhat in 
diameter, leaving an annular dark space between it and the 
glass. The Faraday space became gradually indistinct over the 
Hg. cathode, and the same phenomenon appeared over the Pt. 
cathode but at a lower pressure. Thus at one stage of the ex- 
haustion the appearances in the two tubes when containing 
the cathode were widely different. At the lowest pressures 
reached, both tubes, so far as clearly seen, were very similar in 
appearance, and the phenomena agreed with those observed when 
both electrodes were of mercury. 

Phosphorescence. 

For clearness let us suppose, as was actually the case, A the 
Hg., B the Pt. electrode. In the tube BO the phosphorescence 
first appeared at the level of the upper portion of the wire, and 
gradually spread both up and down as the exhaustion proceeded, 
reaching the summit G of that tube. In the tube AE the 
phosphorescence hardly appeared within 2 cm. of the Hg. surface. 
In the lower parts of both tubes phosphorescence was most 
brilliant at about the positions of the Faraday spaces at the 
lowest pressures, and so at a much higher level in AE than in 
BO. Also the phosphorescence at E was much more intense 
than that at G, though the latter was very well marked. 

At the lowest pressures reached a faint nebulosity, presumably 
the positive column, extended to a considerable distance along the 
tube GHP. It was difficult to detect when the platinum Avas 
cathode, but when the mercury was cathode it could be traced 
almost as far as the pump. 

There then appeared at H a patch of phosphorescence on the 
convex side of the bend. This was very faint when the platinum 
was cathode, but when the mercury was cathode it was fairly 



236 Mr GJiree, On some experiments on [June 1, 

bright. This is clearly a variety of the phenomena observed by 
Goldstein and by Spottiswoode and Moulton, but it seems worth' 
noticing as the tube GBP did not lead to an anode. The Hg. 
column in the pump did not lead to earth, and further the 
luminosity in BP decreased as the distance from H increased, 
which it would hardly have done if the pump had acted as anode. 

Deposit on the tubes. 

In the tube BG containing the platinum there were two, 
generally distinct, principal areas of deposit on the glass. When 
the platinum had served for some time as cathode there was a 
dense black deposit from a little above the level of the top of 
the wire downwards, and a second less dense deposit separated 
in general from the lower by an almost perfectly clean and 
sharply denned area only about 1 mm. broad but extending 
right round the tube. Roughly speaking, the upper deposit ex- 
tended to about the highest level attained by the extremity of 
the Faraday space, but there were traces of it further up the 
tube. Thus the portions of the walls of this tube where the 
deposit was thickest were precisely those where the phosphor- 
escence was strongest ere the darkening of the glass reduced 
its brightness. At the same time phosphorescence was con- 
spicuous on the narrow patch of clean glass separating the two 
principal areas of deposit, and also in the upper portion of the 
tube FG where no deposit was seen. 

In the tube AE the permanent deposit extended in general 
from 20 or 30 to 80 or 90 mm. above the Hg. surface, none ap- 
pearing in the neighbourhood of E. It was nowhere so thick as 
that on the other tube. Looking at it in strong light one could 
see small drops of mercury scattered about, but its exact com- 
position was not determined. There were other deposits of a 
more temporary character near the Hg. surface. One had the 
appearance of dew spreading up the tube for a few millimetres 
when the Hg. was made cathode, and gradually creeping down 
when the current was stopped, taking only a short time to dis- 
appear. It was seen only at low pressures. In some cases 
there was a sort of white deposit separated from the Hg. surface 
by a clear space of a few millimetres, which increased apparently 
as the exhaustion proceeded. On readmitting air to the tube 
this last form of deposit in great measure disappeared. What 
has been called above the permanent deposit had a blackish 
colour and was but little affected by the readmission of air. 
These deposits by obscuring the portions of the tube where the 
Faraday and Crookes' spaces had to be looked for, increased very 
much the difficulty of reaching final conclusions as to the existence 
of these spaces at the lowest pressures. 



1891.] liquid electrodes in vacuum tubes. 237 

Electrodes Hg. and Al. 

The aluminium electrode was a flat circular plate of about 
two-thirds the internal diameter of the tube. The following ob- 
servations were taken on several occasions: 







Table 


IV. 








Hg. Cathode 






Al. Cathode 




Length of 

Crookes' 

space. 


Distance 

above 
cathode of 

top of 
negative 

glow. 


Distance 

above 

cathode of 

head of 

positive 

discharge. 


Length of 

Crookes' 

space. 


Distance 

above 
cathode of 

top of 
negative 

glow. 


Distance 
above 

cathode of 
head of 
positive 

discharge. 


'5 




18 






11 




8 


20 




5 


12 





11 


22 




7 


13 


1 


15 


23 




12 5 


16-5 


•75 




24 
25 
30 






18 
17 
20 


1-5 




40 






325 


2 




40 
40 






30 
25 


8 




80 


6 




70 



The distances are in millimetres, and were all measured from 
the centre of the cathode surface. At the higher pressures the 
upper limit of the negative glow was pretty distinct and so has 
been recorded above. Observations in the same horizontal line 
were taken in rapid succession as in the case of Table in. The 
ratio of the mean distance from the cathode to the end of the 
positive column when the cathode was mercury to the corre- 
sponding mean distance when the cathode was aluminium is as 
1"37 : 1, and so is very much less than the ratio found when the 
solid electrode was platinum. Since no such striking difference 
between the metals platinum and aluminium as electrodes seems 
to have been noticed by previous observers, the inference would 
seem to be that we are here concerned with the size and shape 
rather than with the material of the cathode. 

Owing to the appearance of a slight crack in the tube with 
the Al. electrode its base had to be immersed in mercury con- 
tained in a paraffin cup. Thus the length of the Crookes' space 
when short could not be accurately observed without lowering 
the cup. This was not done in the cases recorded in the table, 
to avoid the risk of leakage, variation of current, etc., but on 



238 Mr Ghree, On some experiments on [June 1, 

other occasions such observations were taken and it was found 
that from the highest pressures where it was visible down to 
the lowest in the above table, Crookes' space appeared slightly 
but distinctly longer over an Hg. cathode than over an Al. 
cathode. 

Appearance of the discharge. 

Under similar conditions the Al. tube was always redder 
than the Hg. tube, though both were at the lowest pressures 
mainly white. When the pressure was reduced to a certain 
stage, the Faraday space over the Hg. cathode became more 
and more indistinct till it seemed to vanish. The Faraday 
space over the Al. cathode was at this stage unmistakeable, 
but at a lower pressure it too eventually became undistinguish- 
able. The way in which the Faraday space over the Hg. cathode 
disappeared was rather remarkable. 

The pressure reached a point at which the appearance in the 
tube was unstable. There might be an unmistakeable Faraday 
space and negative glow, and then a sudden transition to a stage 
in which distinct striae reached down the tube to near the Hg. 
surface. During this time the discharge in the Al. tube showed 
no fluctuation. As the pressure was carried lower the striae 
became less and less distinct until the Hg. tube whether the 
Hg. were cathode or anode seemed an almost uniform white. 
The lowest 5 or 6 mm. in the tube appeared sometimes per- 
ceptibly, sometimes doubtfully darker than the rest. Distinct 
striae eventually ceased to appear even in the Al. tube, but it 
showed to the end an unmistakeable Crookes' space which how- 
ever was becoming increasingly indistinct at the lowest pressures 
reached. 

The greatest length reached by the Crookes' space over the 
Al. cathode was 14 mm. At this stage the tube containing 
the Hg. cathode was as bright as anywhere within 7 or 8 mm. 
of the mercury surface. 

Phosphorescence. 

When the aluminium was cathode phosphorescence extended 
at the lowest pressure throughout the whole of the tube BG, 
whereas when the mercury was cathode phosphorescence was 
not observed within some 2 cm. of its surface. The phosphor- 
escence at the top E of the Hg. tube was always brighter than 
that at G at the same exhaustion. At the best exhaustion with 
the Hg. cathode there was distinct luminosity along the tube 
GHP as far as the pump, a distance of 42 cm. from H, and a 
patch of phosphorescence was distinctly visible at H. At the 



1891.] liquid electrodes in vacuum tubes. 239 

same exhaustion with the Al. cathode a faint luminosity was 
seen to some distance past H, and a very faint phosphorescence 
could be made out at the bend. 

Deposit on the tubes. 

In the Hg. tube there were the same deposits as before, and 
a further one was observed under the following circumstances. 

On several occasions on examining the Hg. surface by day- 
light it was found to present a yellow metallic appearance. The 
conditions preceding its first appearance were as follows. An 
air-bubble which had remained under the mercury was driven 
up whilst the discharge was passing. Mercury splashed up the 
walls of the tube and adhering to some extent presented a 
concave surface. Some observations were taken with the surface 
in this state, and next day it was noticed that the surface was 
yellowish, and that there were traces of a yellow deposit not 
only up the tube AE but also at intervals along DF and even 
for some distance down FB. 

On another occasion when the tube had just been carefully 
cleaned and dried and fresh mercury introduced, I noticed after 
passing the spark both ways for some time that the Hg. surface 
though retaining its ordinary convexity had a decidedly yellow 
appearance, and that the tube near it was slightly yellow. Air 
was allowed to leak in, and the tube being left for some days 
appeared when next examined quite clean, while the Hg. surface 
had its usual colour. When however the tube was again ex- 
hausted and left for some days, the Hg. surface and a few 
millimetres at the base of the tube were found yellow as before. 
However, on passing the spark the phenomena had their normal 
character — which was not the case when the mercury surface 
was concave — and on re-examining the tube the yellow colour 
was found to have entirely disappeared and it was not observed 
again. With the exception of the yellow patches above men- 
tioned and a narrow dark ring sometimes observed on the glass 
near the head of the positive column, the tube with the Al. 
electrode showed no distinct deposit. The origin of the yellow 
colour was not discovered. Its appearance in the tube FB sug- 
gests but does not prove a capacity in the discharge to transport 
particles from a cathode surface round corners. The transporting 
agency might of course have been vapour rising from the Hg. 
surface and condensing on cooler portions of the tube. 

Effect of sudden alteration of the Hg. surface. 

The experiments on this point were those first carried out. 
Both electrodes were then of mercury, and the tube GHP instead 



240 Mr Chree, On some experiments on [June 1, 

of being fused to the pump as it was during the other experi- 
ments, was connected to it by some thick-walled india-rubber 
tubing. The mode of altering the surface was simply by shaking 
or vigorously tapping the tube. With the head of the positive 
column from 9 to 30 mm. above the cathode, the stage at which 
the Faraday space was most distinct, no certain change in its 
position was observed to follow the alteration of the surface. 

The only stage of exhaustion at which a distinct effect of any 
kind was observed was that where the Faraday space seems to 
be in the act of disappearing. On first starting the current, 
especially with a slow break and noisy spark, there appeared 
more or less uniform whiteness in the cathode tube and in- 
distinct striae in the anode tube. After the discharge had 
passed a short time, the colour tended to fade out of a portion 
of the cathode tube., roughly speaking, between 30 and 90 mm. 
over the Hg. surface, and phosphorescence became much more 
conspicuous, especially in this part of the tube ; also the striae 
in the anode tube became more distinct. A shaking or sharp 
tapping of the tube instantly restored the more or less uniform 
white colour throughout the cathode tube and tended to obliterate 
the striae in the other tube. The effect lasted only a short time, 
the discharge gradually reverting to the appearance it presented 
before the disturbance. This phenomenon invariably presented 
itself under the conditions stated. The only explanation that 
occurs to me — suggested by the views of Messrs E. Wiedemann 
and H. Ebert — is that, at least at certain stages of exhaustion, 
the condition at the cathode surface which leads to the pro- 
jection of molecular streams takes some time for its full develop- 
ment, and that on its development the successive discharges 
follow one another more rapidly and consist each of a smaller 
quantity of electricity. The shaking of the tube and consequent 
distribution of fresh mercury over the cathode surface restores 
the original conditions, which are less favourable to the production 
of molecular streams. 

Electrodes H 2 S0 4 and Al. 

In the next set of experiments the electrode in AE was some 
pure sulphuric acid, the aluminium plate electrode in the other 
tube being retained. 

Some experiments with sulphuric acid electrodes have been 
described by Paalzow 1 . He gives an interesting account of the 
electrolysis of the acid, and of spectroscopic observations on the 
discharge. He observed the positive discharge to start from the 
line of separation of the fluid surface and the wall of the tube. 

1 Wied. Ann. 7, 1879, pp. 130—135. 



1891.] liquid electrodes in vacuum tubes. 241 

As I hardly follow his description of the appearances at the 
cathode I give his own words : " Von der negativen Fliissigkeits- 
oberflache selbst erhebt sich in einigem Abstande von derselben 
ein schwach conischer Lichtring, ahnlich wie die Flamme eines 
ringformigen Brenners," p. 131. With increasing exhaustion : 
" um so mehr verlangert sich dieser negative Lichtcylinder, und 
ura so grosser wird sein Abstand von der Flrissigkeitsober- 
flache," p. 132. At very low pressures he observed the pheno- 
mena to be much the same at both electrodes. 

In this case my observations commenced as soon as the 
pressure was sufficiently reduced for the discharge to become 
visible in a dim light. This invariably occurred when the alu- 
minium was cathode, and the luminosity took the form of a 
thin purplish negative glow. At somewhat lower pressures the 
Al. tube was fairly luminous whether it contained the cathode 
or anode, the H 2 S0 4 surface when cathode showing a thin blue 
glow. The horizontal tube and a small portion of the tube AE 
below D then showed a red spark discharge. The greater portion 
of the latter tube remained however dark, except that at in- 
tervals red twig-like discharges passed down it. At this stage 
the spark in the tube DF was most twig-like and of least diameter 
at that end which was nearest the cathode, whether Al. or H 2 S0 4 . 
The pressure had to be further reduced to a considerable extent 
before luminosity could be detected at the H 2 S0 4 surface when 
anode. When the discharge was first clearly seen at an H 2 S0 4 
anode it took the form observed at pretty low pressures with 
an Hg. anode. Throughout the greater portion of the tube the 
positive column appeared as a solid cylinder of considerably less 
diameter than the interior of the tube, but near its base this 
cylinder increased in diameter so as just to fill the tube on 
reaching the liquid surface. 

At this stage the appearance in the tube AE over an H 2 S0 4 
cathode took the following form. A column of very small 
diameter, usually bright red in colour, extended down the axis of 
the tube to within a short distance of the liquid surface. The 
end of this column was sometimes truncated, but frequently it 
presented a sharp point like that of a pencil. Over the liquid 
surface, completely occupying the cross-section of the tube, there 
existed the ordinary blue negative glow. At first this was sepa- 
rated by several millimetres of a dark, presumably Faraday, space 
from the red column, but as the exhaustion proceeded while the 
head of the column retired from the H 2 S0 4 surface the negative 
glow overtook it. With progressing exhaustion the point and a 
gradually increasing length of the red column were immersed in 
the blue glow. The difference in colour rendered the red column 
conspicuous through the blue glow, but I am unable to say 



242 



Mr Ghree, On some experiments on 



[June 1, 



whether the two were actually in contact. Possibly they may 
have been separated by a thin hollow conical dark space. 

At this stage the tube containing an Al. cathode showed a 
positive column whose diameter was several times greater than 
that of the column just described. Further, the head of the 
positive column over an Al. cathode had the usual convexity, and 
was separated by a distinct Faraday space from the negative glow. 
As the exhaustion proceeded an ordinary Crookes' space appeared 
over an Al. cathode, and the other phenomena seemed of the usual 
type. Over an H 2 S0 4 cathode, however, the phenomena seemed 
of an exceptional kind. 

The liquid surface was of course concave, the depression of 
the vertex below the rim being about If mm. Thus on looking 
sideways at the tube one saw a curved dark line answering to 
a vertical section of the liquid surface. Between this and the 
horizontal plane through the rim of the surface there appeared 
a blue tint. This might have proceeded from a luminous film 
over the liquid surface or merely from the reflexion of the glow 
further up the tube. Now supposing a Crookes' space to exist, 
the surface separating it from the negative glow would by analogy 
from other cases be expected to resemble in form the liquid 
surface. Thus so long as the thickness of this space was less than 
the sagitta of the liquid meniscus, one would have expected the 
blue negative glow to pass near the axis of the tube into the 
blue of the liquid surface, while close to each of the tube walls 
one would look for a dark space shaped like a right-angled 
triangle, with a curved hypothenuse answering to the surface 
of the negative glow. Let us call this type (a). An idea of 



(6) 

C 



I) 



(<0 

C 



Fig. 2. 



B 



E 



it will be easily derived from (b), fig. 2, by supposing ACB non- 
existent. The spaces HAD and KBE are to be supposed black, 
while below DABE and above HABK blue prevails for some 
distance. DABE represents the liquid rim cutting the surface 
HABK of the negative glow in the straight line AB. At lower 
pressures one would have expected to see the length AB 
gradually diminish, till finally there appeared over the whole 



1891.] liquid electrodes in vacuum tubes. 243 

liquid surface a black space, which would appear to the eye to be 
bounded below by a straight line and above by the curve of 
intersection of the negative glow by a vertical plane. Let us 
call this type (d). These types were actually observed, but in 
addition two other types were seen, viz. those represented by (b) 
and (c), fig. 2. 

In both, BABE represents the rim of the liquid surface as 
seen by an observer's eye at the same level. In (6) the small 
areas HAD, KBE had a black appearance, but were in general 
not conspicuous. The negative glow extended from HABK for 
some distance up the tube as in type (a). But it now showed a 
sort of tuft ACB of a much whiter and less translucent blue than 
the rest, with more or less distinct curvilinear boundaries which 
seemed to cross at G and gradually fade away beyond it. In (c) 
there was no trace of any black spaces, and the tube appeared 
prevailingly blue all the way up with the exception of an almost 
pure white tuft ACB. It had a tolerably distinct outline except 
at the top C where the colour passed gradually into blue. It was 
sometimes more conical and sometimes more depressed seemingly 
than in the figure, but exact measurements were not attempted. 

A slight unsteadiness in the type (c) happened to catch the 
eye when looking at the liquid surface while putting on the 
current. Further examination led to the following results. 

After keeping the pump at work for some time and then 
suddenly starting the discharge with the H 2 S0 4 as cathode one 
saw a sort of white cloud instantly gather in the tube, separated 
apparently from the liquid surface by from 2 to 4 mm. of a 
dark interval. The cloud, however, immediately stretched down 
the tube and transformed into the appearance (c). This trans- 
formation was on several occasions so distinctly seen that the 
observer could hardly be mistaken ; but it was not always seen 
under these conditions. The whole thing happened so fast that 
sometimes all one could say was that some rapid change had 
occurred. When the current was simply stopped and renewed 
without intervening exhaustion, the tuft appeared at once in the 
position shown in the figure. 

I am not aware of previous observations on the form of the 
negative glow near concave cathodes forming surfaces of revolu- 
tion, and it would obviously be very difficult to see what actually 
exists within the rim of such a cathode. Professor Crookes, how- 
ever, in Plate 14, Phil. Trans. 1879, gives some beautiful coloured 
illustrations which show the negative glow and the Crookes' space 
near a concave cylindrical cathode. I would more especially draw 
attention to his illustrations b and c fig. 11, which show a great 
concentration of negative glow near the plane of symmetry 
through the axis of the cathode. In Crookes' fig. b there is a 

VOL. VII. pt. IV. 19 



244 Mr Chree, On some experiments on [June 1, 

regular tuft, and in his figure c the appearance of a bundle of rays 
projecting into a dark Faraday space. In fact if in these figures 
all to the left of the plane through the rim of the cathode were 
removed there would be a considerable resemblance to the pheno- 
mena illustrated by my (b) and (c) fig. 2. The principal difference 
is that with Crookes' electrode there appears to have been a 
Faraday space limiting a distinct and, except in the plane of 
symmetry, thin negative glow, whereas with the H 2 S0 4 cathode 
no such distinct Faraday space was seen. 

I would also call attention to Crookes' fig. 2, p. 643, Phil. 
Trans., 1879, as showing a concentration of negative glow in 
positions opposite the hollows of a corrugated cathode. 

The resemblance of the boundaries of ACB in (b) and (c) 
fig. 2 to caustic surfaces unquestionably suggests that we may 
have here to do with some species of emission from the surface 
separating the liquid and gas, each element of the surface acting 
as a source more or less independent of its neighbours. If, as 
seems to be the case with the molecular streams 1 , the direction of 
emission from the elements near the rim deviates more than else- 
where from the normal to the surface, one can easily see that there 
would be a crossing of the trajectories near the axis of the tube, 
even close to the liquid surface. 

The sudden change noticed when the type (c) fig. 2 appeared, 
might be accounted for by the discharge leaving at first from only 
a portion of the liquid surface, or possibly it may be connected 
with the rise of the small bubbles accompanying the electrolysis 
which take some short time to reach the surface. 

The following table gives results from several series of observa- 
tions, all data in a horizontal line being taken in immediate 
succession with a constant rate of make and break. All the dis- 
tances are in millimetres. Except at the highest pressures even 
the approximate position of the upper end of the negative glow 
could not be fixed. Such an entry as "7 +" in the column headed 
"Distance... glow" means that the glow reached to some unde- 
termined height above the point of the red column. In the case 
of the H 2 S0 4 cathode all the distances were measured from the 
rim as the most convenient starting-point. It ought to be remem- 
bered, however, in instituting any comparisons that in the axis of 
the tube the true liquid surface was some If mm. below the rim. 
The negative glow in type (d) had a considerably greater curva- 
ture than the liquid surface, so that its distance from that surface, 
measured parallel to the axis of the tube, was least in the axis. 

1 See Goldstein, Wied. Ann. 15, 1882, pp. 254—277, specially pp. 274—5. 



1891.] 



liquid electrodes in vacuum tubes. 



245 



Table V. 



Crookes' space 

^__ X 

Type Height 
above 
rim. 



(a) 



(b) 

C°) 

(d) 

(d) 

(d) 
(d) 



H 2 S0 4 cathode 

Distance 
above 
rim of 
top of 

negative 
glow. 

4 

6 

7 + 
7 + 
13 
10 + 
13 + 
15 + 



Al. cathode 



1-5 
1-5 

2 
2-5 



17 + 

18 + 
24 + 
30 + 



Distance 

above 

rim of 

head of 

positive 

discharge. 

5 

6 

7 

7 

10 
10 
13 
15 
absent 
absent 
17 
18 
24 
30 



Length 

of 

Crookes' 

space. 



Distance 

above 

cathode 

of top of 

negative 

glow. 



17 



2-5 
4 

4'5 

5 

5 

7 
8 



23 



Distance 
above 

cathode 
of head of 

positive 
discharge. 

11 

17 
19 
22 

29 

40 
38 



dubious 

50 
dubious 
dubious 



None of the distances admitted of very accurate observation 
because the positions of the various parts of the discharge, especially 
the head of the positive column over an Al. cathode, were seldom 
quite stationary. At the lower pressures in the table the blue 
light surrounding the red column over an H 2 S0 4 cathode extended 
up the tube AE to above the level of the horizontal tube. When, 
however, the interrupter was adjusted to give a slower and more 
noisy spark the red column sometimes completely disappeared. 
It also sometimes faded out when the discharge was kept passing 
for some time. In either case there appeared from about 20 to 
40 mm. above the H 2 S0 4 cathode a considerably darker blue than 
elsewhere verging towards black. This darker band always accom- 
panied the types (6) and (c) of discharge with which the red 
column was not observed. Once or twice this column instead of 
being red w r as white. In general its outline appeared straight and 
regular, but on at least one occasion it showed numerous short 
horizontal projections like hairs. 

The colour of the positive column at the higher pressures and 
until the Crookes' space over the Al. cathode attained a length of 
1 or 2 mm. was always red. At lower pressures when red its 
colour was much fainter, and at the lowest pressures it generally 
tended to white or even blue. The striae were most conspicuous 
when the colour w r as white, but they were seldom very distinct. 
At the lowest pressures only faint phosphorescence was observed in 

19—2 



246 Mr G. H. Bryan, Note on a Problem [June 1, 

the tube over an Al. cathode, and none was noticed overanH 2 S0 4 
cathode. There was also no deposit on the walls of the tube. 
The joint absence of these two phenomena is rather striking, but 
the exhaustion was not, I think, carried so low as in the previous 
experiments. 

At the lowest pressure attained, with a Crookes' space of from 
9 to 10 mm. in length over an Al. cathode, the appearance of the 
discharge became unsettled. There was nowhere any very bright 
colour, but throughout the greater part of the tube over an H 2 S0 4 
cathode the colour along the axis was unmistakeably red. There 
was a red column somewhat resembling that seen at higher 
pressures, terminating in a sharp point about 40 mm. over the 
HoS0 4 surface. Immediately below this the tube appeared blue 
throughout the entire cross section, but a little lower down there 
appeared a red column of about half the diameter of the tube. Its 
base was curved, and its convexity was directed towards the 
H 2 S0 4 surface, from which it was separated by an interval of only 
2 to 4 mm. The colour of this interval seemed to vary from black 
to faint blue. Both these red columns had a blue haze between 
them and the walls of the tube. This stage seems to answer to 
that observed with a mercury electrode previous to the discharge 
assuming a nearly uniform appearance throughout. 

My best thanks are due to Professor Thomson for putting at 
my disposal the necessary apparatus, and for numerous suggestions 
during the course of the experiments, which were performed in the 
Cavendish Laboratory. 

(5) Note on a Problem in the Linear Conduction of Heat. By 
G. H. Bryan, M.A., St Peter's College. 

The problem of conduction of heat in a bar one end of which 
is subject to radiation while the other end is at an infinite distance 
away, has been treated by Mr Hobson in his paper "On a Radiation 
Problem " published in the Proceedings, Vol. VI., page 184. The 
author there finds the expressions in the form of definite integrals, 
representing the temperature due to the given distribution of heat 
in the medium at the extremity of the bar and to the initial 
distribution of heat in the bar respectively. 

The expression which Mr Hobson obtains for the second part of 
the temperature is, however, open to several objections. It appears 
to fail entirely if the initial temperature is anywhere discontinuous 
or if any sources, doublets, or other singularities are initially present 
in the bar. 

Moreover, from the integral obtained, it is shown that the 
initial distribution is equivalent to a certain distribution of lines 
of sources and sinks in a rod extending to infinity in both directions, 



1891.] in the Linear Conduction of Heat. 247 

but this interpretation is also liable to objection. For the distri- 
bution of sources and sinks corresponding to the initial temperature 
in any single element of the rod is not in itself equivalent to that 
element. Hence Mr Hobson's solution gives no idea of the part 
played by the initial temperatures of the separate elements of the 
rod, and in fact it does not even give a correct result if the effect 
of the initial temperature in any part of the rod is considered 
apart from that in the rest. 

The problem is best solved by starting with a single instan- 
taneous source at one point of the rod, since from this the more 
general solution can be obtained by integration. 

Consider then the effect of an instantaneous source Q of heat 
generated at the point x at the time 0. The temperature due to 
such a source if there were no boundary would be 

Q {x-xj 

exp — 



*Jife * 4ft " 

Let this expression be denoted by v 2 , and let the temperature 
when the boundary is taken into account be v where 

v = v 1 + v 2 . 

If the external medium be at temperature zero, the boundary 
condition gives, when x = 0, 

dv 7 n 
ax 

„ dv., , (d ,\ Q (x-x'f 

Hence -y- 2 — hv = - -^ — h) 9 ,— r exp - 



dx \dx J^Jirkt 4ft 

Q f x' ) x' 2 

= 2jm I m + h \ exp - m when ■ ■ °- 

But v 2 must be the temperature due to a series of images on 
the negative side of the origin, hence the conditions of the problem 
will be satisfied by taking 

dv., 7 Q f on + x' , 1 (x + x'f 

^"^ = 27sr"w- +A r xp --4ft- 

_(d ,\ Q (x + xj 

"U + Jvs exp T" 

_fd \ Q (x + x'Y 

'{dx'VYjm^ ^ 

2hQ (x + x'f 

+ 2 v / 7r ft eXP 4ft ' 



248 Mr 0. H. Bryan, On Conduction of Heat. [June 1, 1891. 

Therefore by integration 

Q (x + x'f . r 2hQ „ (1 4- a/)" 7f . 

In the last integral put g — oc + z. Also add fl 1} thus the whole 
temperature assumes the form 

Q f (x-aTf (x + x'f) 

v = v.+v= ^— y=^exp - , T . + exp- V 



-2/i( 



Qe~ fe (m + z + aff , 
- — . exp — . ,- — - . dz. 



The first line represents the temperature due to the given source 
and an equal source at the geometrical image, the second line 
represents the temperature due to a line of sinks extending from 
the geometrical image to infinity in the negative direction. None 
of the images of the source lie on the positive side of the origin. 

For the temperature due to the initial distribution v = <$>(x) we 
write (p(x')dx' for Q, and deduce by integration, 

1 r\ {x-xj (x + xj) ,,,.,, 

2h r 00 / -00 (x + x'f 

" zjm J. I, exp (_ fe) • exp — m * w **• 

This solution holds good whether <f> (x) be finite and continuous 
or not. 

[Note by Mr Hobson. — Mr Bryan's formula is undoubtedly an 
improvement on the one I gave in the paper referred to. It should 
be observed that his formula may be obtained from mine by 
integration by parts.] 



PROCEEDINGS 



OF THE 



Camhriirge lIjHnsq^tnd Bmtty. 



October 26, 1891. 

ANNUAL GENERAL MEETING. 
Professor G. H. Darwin, President, in the Chair. 

The following Fellows were elected Officers and new Members 
of Council for the ensuing year : 

President : 
Prof. G. H. Darwin. 

Vice-Presidents : 
Prof. Hughes, Prof. Thomson, Mr J. W. Clark. 

Treasurer : 
Mr R. T. Glazebrook. 

Secretaries : 
Mr J. Larmor, Mr S. F. Harmer, Mr E. W. Hobson. 

New Members of Council: 
Mr H. F. Newall, Mr C. T. Heycock, Mr A. E. H. Love. 

The following Communications were made to the Society : 

(1) On the Absorption of Energy by the Secondary of a Trans- 
former. By Prof. Thomson. 

vol. vii. pt. v. 20 



250 Mr Sedgiuick, on a Peripatus from Natal. [Nov. 9, 

(2) On an experiment of Sir Humphry Davy's. By G. F. 0. 
Searle, M.A., Peterhouse. 

Two copper wires are passed up through holes about 5 centi- 
metres apart in the bottom of a flat trough, their ends being level 
with the surface of the trough. Mercury is then poured into the 
trough to a depth of about 4 millimetres. On sending a powerful 
current through the mercury by means of the two wires the 
mercury in the immediate neighbourhood of the electrodes was 
elevated into a small cone 2 or 3 millimetres in height. 

(3) Some notes on Clark's Cells. By R T. Glazebrook, M.A., 
Trinity College, and S. Skinner, M.A., Christ's College. 

The paper relates to the causes of the variation of electromotive 
force of the cells. In addition to the causes indicated by Lord 
Rayleigh the authors find that the state of amalgamation of the 
zinc pole may cause a fall in force if the zinc does not shew a 
bright surface. This is worked out by means of a testing cell into 
which the faulty zincs are transplanted. The result is confirmed 
by Swinburne's experiments on zinc rods in zinc sulphate solution. 
To correct this fault previous amalgamation in the presence of 
dilute sulphuric acid is recommended, or immersion of the zinc in 
the paste. Dr Hopkinson's method of testing cells by tapping 
was shewn. 

(4) Illustrations of a Method of Measuring Ionic Velocities. 
By W. C. D. Whetham, B.A., Trinity College. 

(5) On Gold-Tin Alloys. By A. P. Laurie, M.A., King's 
College. 

November 9, 1891. 

Dr Gaskell in the Chair. 

The following Communications were made to the Society : 

(1) Note on a Peripatus from Natal. By A. Sedgwick, M.A., 
Trinity College. 

[Received November 10, 1891.] 

Last spring I received from Mr P. S. Sutherland a single 
specimen of Peripatus, which was found by Mr J. F. Quickett in 
the Botanic Gardens of Pietermaritzburg, Natal. The specimen 
possessed 22 pairs of claw-bearing legs, and presented all the 
characters of P. Moseleyi, as described in my monograph of the 
genus. 



1891.] Mr Bateson, On Variations in colour of cocoons. 251 

The specimen measured h\ centimetres in length. The ventral 
surface was of the usual light colour. The ground colour of the 
dorsal surface was dark green, with a sufficient number of yellow 
papilla? to give a yellow tinge to the specimen. There was a light 
yellow band, from which the green was almost entirely absent, just 
dorsal to the insertion of the legs. 

The specimen was a female, and the last leg was without a 
white papilla. 

The genital opening was subterminal, and behind the legs of 
the last pair. 

The uterus was full of embryos, at a stage of development 
more advanced than in a Cape specimen killed at the same time of 
year. 

The number of legs of the embryos was, as in the adult, 22 
pairs. 

The specimen is of interest, as being the first recorded from 
Natal. 

(2) On Variations in the Colour of Cocoons (Saturnia carpini 
and Eriogaster lanestris), with reference to recent theories of Pro- 
tective Coloration. By W. Bateson, M.A., St. John's College, 

[Abstract; received November 11, 1891. Reprinted from the Cambridge 
University Reporter, November 24, 1891.] 

The cocoons of several moths, e.g. the Emperor and Small 
Egger, vary in colour from dark brown to white. It is believed by 
some that these colours have a protective value as a means of con- 
cealment, and it has been stated by Poulton and others that when 
spun on leaves which will turn brown, or in dark suiTOundings, the 
cocoons are dark, while they are white if spun on white paper. To 
account for this phenomenon " the existence of a complex nervous 
circle" has been assumed. The present experiments shewed that 
it is true that larvas left to spin on their food-leaves produce dark 
cocoons, and also that if they are taken out and put in white paper 
the cocoons are white. But it was found that larva? similarly 
taken out and made to spin in dark substances also spun white 
cocoons, and indeed that starvation, or merely interference at the 
time of spinning, may lead to the production of a white cocoon. On 
the contrary, if white paper is put amongst the food, so that the 
larva? can, of their own choice, walk into it and spin, the cocoons 
are generally dark. It was noticed in several cases that larva? 
which had been shut up evacuated a quantity of dark juice having 
the natural tint of the cocoon, and the suggestion was hazarded 
that absence of colour in the cocoon perhaps results from the loss 
or retention of this juice, which may be of the nature of me- 
conium. 

20—2 



252 Miss R Alcoclc, On the Digestive [Nov. 9; 

(3) Exhibition of Phylloxera vastatrix. By A. E. Shipley, 
M.A., Christ's College. 



Professor Hughes in the Chair. 

(4) The Digestive Processes of Ammocoetes. By Miss R. 
Alcock (communicated by Dr Gaskell). 

[Received December 3, 1891.] 

In all the higher vertebrates digestion is carried on by means 
of the secretion of specialised glands localised in certain definite 
portions of the alimentary canal or of glandular masses which are 
formed as appendages of the same. The formation of a peptic fer- 
ment is confined to the glands of the stomach, of a tryptic ferment 
to the pancreas and of diastatic ferments to the salivary glands and 
pancreas. Passing to lower forms we find in Amphibia that the 
formation of peptic ferment is not restricted to the stomach, but 
extends oralwards into the oesophagus, which is even more active 
as an organ for digestion than the stomach. Then in Fishes this 
tendency to diffuseness in the position of the proteid-digesting 
glands is even more pronounced, and it is remarkable, as Kruken- 
berg has pointed out, how their position varies even in nearly 
allied families. Sometimes a pancreas is present, sometimes 
absent; in some the appendices pyloricae are well developed, in 
others they do not exist, and in some they function as digestive 
glands, whilst frequently they seem merely to act as organs of 
absorption. In some cases the so-called pancreas does not func- 
tion in proteid digestion, and in many fishes certain cells forming 
part of the liver secrete a tryptic ferment, which enters the ali- 
mentary canal by means of the bile-duct. The pepsin-forming 
glands also vary in position, sometimes extending oralwards and 
sometimes into the upper part of the intestine. 

These observations of Krukenberg lead to the conclusion that, 
as far as digestive organs are concerned, the lower we descend in 
the scale of evolution of the vertebrates, the greater is the ten- 
dency towards a decrease of specialisation in function and a diffuse- 
ness in the position of the secreting cells. If this is the case, the 
study of the digestive processes in the lowest vertebrates ought to 
shew this absence of concentration of the secreting tissues in a still 
more pronounced manner. With this object Dr Gaskell proposed 
to me to find out how the digestive processes were carried on in 
the Ammocoetes, as no physiological observations have been made 
on the digestion of this primitive vertebrate by Krukenberg or any 
other observer. 



1891.] Processes of Ammocoetes. 253 

We may consider for this purpose that the alimentary canal 
consists of three portions, 1st the pharynx, 2nd the narrow tube or 
anterior intestine which leads from the pharynx to the mid-gut 
and terminates posteriorly at the entrance of the bile-duct where 
the sudden enlargement of the alimentary tube marks the begin- 
ning of the 3rd portion, the intestine proper. The glandular 
appendages in connection with these parts are, (1) the so-called 
thyroid gland, with its duct opening into the pharynx, and (2) 
the liver with its duct opening into the intestine. 

In order to test for the presence of digestive ferments, I made 
extracts of the epithelium lining the pharynx, the liver, the intes- 
tine and the thyroid in *2 °/ HC1. or in glycerine, using in each 
experiment the organs of two or more Ammocoetes of Petromyzon 
Planeri. I may here mention that I can confirm Krukenberg's 
experiments on the digestive ferments in fishes and invertebrates 
as to the temperature at which they are most active. I find my 
extracts are very much more powerful at 38° to 40° C. than at 15° 
to 20° C, many authors giving the lower temperature as that at 
which the digestive extracts of fishes and cold-blooded animals 
generally are most active. I found that all parts of the alimentary 
canal, with the exception of the thyroid gland, were capable of 
digesting fibrin in a 2 °/ HC1. medium with greater or less 
rapidity. I used carmine-stained fibrin after Griitzner's method, 
and always used as control experiment a *2 °/ o HC1. solution alone 
and an extract of some tissue of the animal which was inactive in 
digestion. 

The results of these experiments may be summed up as 
follows : — The extract of the liver is always the most powerful ; — 
thus in one case about 1 c.c. fibrin was entirely digested in from 
2 to § hr. by an extract made from the livers of two Ammocoetes. 
The epithelium of the pharynx comes next in activity, thus in the 
case mentioned the epithelium of the pharynx of the same two 
animals digested the same amount of fibrin in about 1^ hrs. Next 
in order of activity comes the extract of the intestine ; this always 
gives decided evidence of digestive power, but if carefully cleaned 
out with a soft brush before the extract is made, so as to remove 
as far as possible the secretion from the liver, then its power of 
digestion is very far behind that of the pharynx or liver, although 
in all cases digestive activity is still manifest. Finally the extract 
of the thyroid is always inactive. 

So far I have not succeeded in obtaining any results in a 1 °/ 
sodium carbonate medium, and conclude that tryptic ferment is 
absent. In relation to this it is interesting that in a youno- 
Selachian Krukenberg found that a tryptic ferment was entirely 
wanting, and he suggests, for this and other reasons, that in primi- 
tive vertebrates the digestion was rather peptic than tryptic. 



254 Mm R- Alcock, On the Digestive [Nov. 9, 

I conclude then from these experiments that : — 

1. The proteid digestive ferment in the Ammoccetes is of the 
nature of pepsin rather than trypsin. 

2. This ferment is very diffuse in position, being found in all 
parts of the alimentary tract. 

3. It is found mainly in the anterior part of the tract, 
especially in the respiratory portion of the pharynx and in the 
liver. 

4. The so-called thyroid gland has nothing whatever to do 
with the digestion of proteids. 

Perhaps the most novel and important feature of this series of 
experiments is the evidence of the importance of the pharyngeal 
cavity for proteid digestion in this primitive form of vertebrate ; 
and when we come to examine the histological structure of the 
alimentary tract we find that glandular secreting structures are 
more conspicuous in this part of the alimentary canal than in the 
intestine proper. In the pharynx the epithelium lining the body- 
wall and the adjacent branchial surfaces is undoubtedly different 
from the single layer of flattened epithelium cells which cover the 
lamellar folds of the branchiae themselves. It is usually described 
as consisting of several layers ; but very conspicuous amongst the 
small epithelium cells are numerous glandular looking cells, to 
which I have never found any reference in descriptions of this 
region. These cells are arranged in groups of five or six together, 
and correspond in height to the whole thickness of the epithelium; 
they are somewhat swollen in the middle and are covered super- 
ficially by the small epithelium cells with the exception of a small 
space above the centre of the group where their tips converge to- 
gether at the surface. In some preparations a collection of deeply 
stained granules can be seen at the outer ends of these cells, and 
in others a stringy mucus-like secretion is issuing from them. It 
is striking how on the branchial folds of the anterior wall of the 
first gill-pouch the epithelium containing these cells predominates, 
only the most internal folds appearing to have retained their respi- 
ratory function. Clearly the abundance and the evident activity of 
these pharyngeal glands is a sufficient histological reason why the 
extract of the pharynx is so active in digestion. On the other 
hand the epithelium of the narrow anterior intestine consists of a 
single layer of tall columnar cells with a distinct cuticular border 
and uniformly ciliated ; there is no histological evidence here of the 
presence of any secreting cells. The epithelium of the rest of the 
intestine is very uniform, and similar in character to that of the 
anterior intestine, though only the anterior dilated portion is 
ciliated ; and the ciliation even here is not uniform, but occurs in 
patches. The whole structure of this region suggests an organ of 
absorption rather than of digestion. 



1891.] Processes of Ammocoetes. 255 

The only glandular organ in connection with the intestine is 
the liver, with its large gall-bladder ; in structure it is typically a 
tubular liver throughout, and there is no evidence of any differen- 
tiation of function in the cells. 

Round the entrance of the bile-duct into the intestine are a 
few small glandular follicles which have received the name of 
pancreas, but in most specimens they are too few and too small for 
it to be possible that they could play any important part in diges- 
tion, at all events during the Ammoccetes-stage. After transfor- 
mation the liver becomes completely separated from the alimentary 
canal, the duct becomes obliterated, and the liver itself undergoes 
fatty degeneration. At the same time the small glands in the 
wall of the intestine which have been called the pancreas increase 
in number and form quite a prominent ring. If these glands are 
pancreatic in function, it suggests itself that possibly the tryptic 
digestion may become more important than the peptic as the 
animal advances to the higher stage of evolution in the adult 
Petromyzon. I hope to be able to work out the digestive processes 
of the adult animal as soon as I can obtain material. 

Finding the presence of a peptic ferment to be so general in all 
parts of the alimentary canal, the question arises if it is present 
also in other tissues of the animal. I have already mentioned that 
no sign of digestive activity can be obtained from extracts of the 
so-called thyroid, extracts of the central nervous system are also 
quite inactive, and extracts of muscle shew only the faintest signs 
of activity after many hours. Then I thought it might be of 
interest to test for peptic ferment in the skin as being an epithelial 
structure continuous with the lining of the pharynx and on account 
of its active secretion ; and I found that a "2 °/ HC1. extract of 
the skins of two Ammocoetes digested 1 c.c. fibrin in a little over 
1 hr. The skin consists of several layers of cells, the most super- 
ficial being those which secrete. They are peculiar in having a 
very thick cuticular border, whose striated appearance is due to 
the presence of fine pores through which the secretion exudes. 
These cells are full of granules and when treated with methylene- 
blue in the living condition, as observed by Mr Hardy, the granules 
at the base of the cell stain blue, whilst those towards the surface 
appear rose-coloured in artificial light ; this reaction he has every 
reason to believe indicates the presence of a zymogen. 

As to the significance of this secretion, it does not seem 
probable that it could function in the digestion of food. I think 
it must have some important function, and considering the habits 
of the animal, which lies buried in the mud, Mr Hardy has sug- 
gested that this secretion of the skin may act as a protection 
against the attacks of bacteria and other organisms, which might 
otherwise be injurious. 



256 Mr Hardy, On the Reaction of certain [Nov. 9, 

(5) On the Reaction of certain Cell-Grannies with Meihylene- 
Blue. By W. B. Hardy, B.A., Gonville and Caius College. 

[Received Nocember 30, 1891.] 

In 1878, Prof. Ehrlich pointed out the fact that the granule- 
containing cells of the body, whether found free in the body fluids 
or elsewhere, could be distinguished from one another by the 
character of the reaction of their granules with different aniline 
dyes*. He distinguished five classes of granules characterised by 
staining with acid, basic, or neutral dyes, or indifferently with acid 
or basic dyes (amphophil). The present communication deals with 
a further subdivision of the basophil granules into two groups, 
characterised by very distinct colour-reactions with the basic dye 
methylene-blue, the one class of granules staining a deep blue, the 
other a bright rose. 

The distinction of tint depends, for some reason not at present 
at all obvious, on illumination with yellow artificial light. Under 
these circumstances the colour-contrast is one of extraordinary 
brilliancy. With daylight, or with gaslight after it has been 
filtered through neutral-tint glass, the rose colour either appears a 
blue like that shewn by the blue-staining granules, or is dulled to 
a blue-violet tint. The explanation of this change may be found 
in the fact that the yellow gaslight is relatively richer in red rays 
(or poorer in blue rays) than is daylight, or the phenomena may be 
of a more complex nature. Be this as it may, the abrupt transi- 
tion from bright rose to bright blue produced by directing the 
mirror from the gas flame to the window is a striking feature of 
this colour-reaction. 

The discrimination of basophil granules into rose-staining or 
blue-staining varieties may depend upon the dichroic nature of 
methylene-blue. Thin films produced by running a minute quan- 
tity of a strong solution under a coverslip appear rose with arti- 
ficial light, while more dilute solutions appear blue. 

Methylene-blue is a salt having the composition of a chloride, 
the base being a pigment of the aniline series, and it has already 
been noticed that alkalies produce a rose or reddish modification 
possibly by decomposing this salt and liberating the pigment-base. 
This suggests that the rose tint may not be solely a physical 
phenomenon but may depend upon a chemical action of the 
granule substance on the dye, or a chemical change produced by 
the osmosis of the pigment through the cell protoplasm to the 
granule. 

That the reaction does not depend simply upon the thickness 

* The various papers dealing with this subject have been republished by 
Br Ehrlich in pamphlet form under the title "Farbenanalytische Untersuchungen 
zur Histologic und Klinik des Blutes." Berlin, 1891. 



1891.] Cell Granules with Methylene- Blue. 257 

of the film of methylene-blue is shewn by the fact that the 
rose-tint may be developed by granules of sizes varying from mere 
points to spherules 2 to 4 \x in diameter. Nor does it depend upon 
the solidity or fluidity of the staining substance; for the contents of 
large vacuoles in the ectoderm cells of Daphnia frequently stain 
an intense rose, the rest of the cell appearing blue. Lastly the 
rose tint may be produced neither in the cell nor in the animal, 
but (in the case of Daphnia) by the action on the dye of a sub- 
stance poured by the ectoderm cells into the surrounding water. 

The hypothesis that the reaction is really of a chemical nature 
is favoured (1) by the general fact that the imbibition of dyes by 
the fresh unfixed cells is determined by the chemical nature of 
those dyes — whether the pigment be basic, or acid ; and (2) by the 
peculiar method of imbibition of dyes by fresh cells. If still living 
cells, such as basophil blood corpuscles, are treated with either an 
organic fluid or normal salt solution, in which a small quantity of 
methylene-blue is dissolved it is noticed that imbibition of the dye 
is coincident with the onset of death. So long as the cell remains 
fully alive it resists infiltration by the pigment, and the granules 
remain uncoloured. This condition may, especially with eosinophil 
cells, last for hours. With the first onset of death the dye makes 
its way through the protoplasm and the granules become coloured. 
Later, when rigor mortis has become thoroughly established, the 
nucleus and cell body absorb the dye, and appear blue. In other 
words the first imbibition of the dye occurs at a period when the 
complex cell protoplasm is commencing to disintegrate, and when 
therefore profound chemical changes are taking place. 

In order to determine whether granules are of the rose or blue 
staining varieties it is necessary to apply the stain in some rela- 
tively innocuous fluid to the living cells; and subsequent treat- 
ment with fixing reagents entirely obliterates the reaction. This is 
because all fixing agents with which I have experimented have 
some action on methylene-blue. Thus corrosive sublimate pro- 
duces a rose-coloured modification, and converts blue staining into 
violet or rose. Ammonium picrate produces a violet tint except 
in the case of very intensely blue granules. Osmic acid converts 
the rose into a blue tint. 

Rose-staining basophil granules have been found by me in free 
basophil cells of Astacus, and of Vertebrates, in the ectoderm of 
Daphnia, and of the Ammoccete larva of Lampreys, and in the 
alveolar cells of salivary glands. 

The last two instances are of a specially suggestive nature, as 
affording instances of cells containing at the same time blue and 
rose-staining granules. In the cells lining the alveoli of the sub- 
maxillary gland of a rabbit I have seen, after treatment with dilute 
methylene-blue in normal salt solution, a zone of rose-coloured 



258 Reaction of certain Cell-Granules with Methylene- Blue. [Nov. 9, 

granules surrounding the lumen and extending about half-way 
towards the basement membrane, while outside this there was a 
zone of blue-staining granules. This suggests that the rose-stain- 
ing condition is a final stage in the elaboration of the constituents 
of the granules of these cells. 

A still more instructive example is found in the ectoderm cells 
of the Ammoccete larva which I examined at the request of Dr 
Gaskell. These cells are each overlaid by a thick cuticle per- 
forated by coarse canaliculi which lead from the cell protoplasm to 
the external surface of the animal. Miss Alcock has shewn that 
these cells, under appropriate stimuli, discharge on to the general 
surface a viscid substance which has the power of rapidly digesting 
fibrin in an acid medium. If these cells are treated with methy- 
lene-blue we find (1) that the extruded secretion gives the rose 
reaction, (2) that the pores in the cuticle may appear as rose- 
coloured rods, owing to their being filled with the secretion, and 
(3) that the cells themselves are occupied by rose-coloured granules 
which lie in the half of the cell next to the cuticular border, and 
by blue-coloured granules which occupy their deeper portions. 

In the ectoderm of Daphnia rose-coloured granules are scanty, 
while, under certain circumstances to be detailed elsewhere, the 
cells may include a number of large vacuoles, the contents of 
which give a brilliant rose reaction. In connection with the 
presence of these vacuoles we find that Daphnia possesses the 
power of extruding on to its surface, through cuticular pores, a 
substance which swells up to form a jelly in water, and stains 
brilliant rose. This particular case will receive more detailed 
description on some future occasion. For the present I will only 
say that the secretion is used by the animal to prevent parasitic 
vegetable or animal growths obtaining a foothold on the shell. 

I have never yet found a blood or lymph cell with both blue 
and rose-staining granules. It may be regarded as probable that 
blue-staining granules are absent from wandering cells. The cells 
with rose-staining granules have a remarkable distribution. In 
Astacus, as I have noted elsewhere*, they occur normally lodged in 
the spaces of a peculiar tissue which forms an adventitia to some 
of the arteries. They are only discharged into the blood as a 
result of special stimuli. In Vertebrates they occur to a marked 
extent in the peculiar adventitia of the blood-vessels of the spleen. 

It is noticeable that I have so far failed to find rose-staining 
granules in endoderm cells, though I have examined the lining cells 
of the alimentary canal and of its glands in very diverse animal types. 

The cells of the excretory organ (end-sac of Daphnia) contain 
granules which have a remarkable affinity for methylene-blue and 
stain a deep opaque blue. 

* Journal of Physiology, 1892. 



1891.] Mr Macdonald, Self-induction of two Parallel Conductors. 259 

November 23, 1891. 
Professor Lewis in the Chair. 

S. Skinner, M.A., Christ's College, was elected a Fellow of the 
Society. 

The following Communications were made: 

(1) The Self-induction of Two Parallel Conductors. By H. 
M. Macdonald, B.A., Clare College. 

(Abstract) 

In Article 685, Maxwell's Electricity arid Magnetism, Vol. II., 

b 2 
the expression ^ (/a -f- //) + 2fi log — , is given for the self-in- 
duction per unit length of two parallel infinite cylindrical con- 
ductors, the radii of their sections being a, a', the distance between 
their axes b, fi, // their magnetic permeabilities, and /n that of 
the suiTounding medium. Lord Rayleigh remarked in the Phil. 
Mag. 1886, that this expression only holds when //, = fx = /jl o . To 
solve the general problem for steady currents it is necessary 
to know the conditions satisfied by the components of vector 
potential at the bounding surface of two media of different 
magnetic permeabilities ; they are found to be 

iap im 7 

<-\ "i" / ^ / ^3 etc. 
/u, on ja on 

Transforming the equations by the relation 
x + ly = c tan \ (£ + trj), 

taking tj - a, w = — /3 as the bounding surfaces of the conductors, 
we have 

d 2 II d"H 4>TTUC0 

d% z drf (cosh t] + cos gy ' ' ' 

d 2 H n d 2 H n n , .. 

d 2 H' d 2 H' ^TTLloy' 

0% drj (cosh 7] + cos I;) 2 ' 

when 7] = a, 

when i] = — /3, 





II- 


= ^0 


and 


1 dH _ 

/J, drj 


1 

^0 






H> 


= H' 


and 


1 

^0 


dH, 

dv 


1 


dH 

drj 


F 


and 6r being 


constant. 











260 



Mr Macdonald, On the Self-induction [Nov. 23, 



Solving these for H, H Q and H' and determining L from them, 
we obtain 



+ ~ 



When fi = fi we have the case of the a conductor iron, the other 
being copper, and then 



aa 



the repulsive force between the conductors being 

fi - fi a' 2 



*efvi- 



fi + fi. b' 2 — a' 



Taking the case of conductors of equal section, the following 
table shews how the variable part of the coefficient of induction 
varies with their distance apart. 



b 


log — -, 
a a 


M + Mo fr-a- 


Increase 
per cent. 


L - 50-5 
Maxwell 


1,-50-5 

from above 


2a 


1-38629 


•282007 


20-3 


2-7725 


3-3365 


3a 


2-19722 


11776 


5-3 


4-3944 


4-6299 


ia 


2-77258 


•06326 


2-2 


5-5451 


5-6716 


5a 


3-21887 


•039829 


1-2 


6-4377 


6-5173 


6« 


3-58351 


•027583 


•7 


7-1670 


7-2221 


7a 


3 89164 


•020211 


•5 


7-7832 


7-8237 


8a 


4-15888 


•015936 


•3 


8-3177 


8-3496 


9a 


4-39425 


•012131 


•2 


8-7885 


8-8127 


10a 


4-60517 


•009851 


•2 


9-2103 


9-2300 



The first column gives the distances between the axes of the 
conductors, the second the values of half the variable term in 
Maxwell's formula, the third half the term which has to be 
added to it, the fourth the increase per cent, of the variable part 
due to the term neglected by Maxwell, the fifth and sixth the 
values of the variable parts in both cases ; /a being taken unity 
and yu. = 100. The table shews that the term neglected is con- 
siderable when the conductors are close to one another, and 
decreases rapidly at first as b increases, afterwards slowly. 



1891. 



of two Parallel Conductors. 



261 



Again taking the conductors touching one another, the follow- 
ing table gives the maximum values of the correction as the 
radius of the iron conductor increases. 





5 


Jog—; 
aa 


""""log,, 6 ' , 

fi + /j. o--a- 


Increase 


L - 50-5 


L - 50-5 from 






per cent. 


Maxwell 


above formula 


a 


2a' 


1-38629 


•282007 


20-3 


2-7725 


3-3365 


2a 


3a' 


1-50407 


•576147 


38-3 


3-0081 


4-1604 


3a 


4a' 


1-67397 


•810307 


48-0 


3-3479 


4-9685 


4a 


5 a' 


1-83257 


1-001419 


54-6 


3-6651 


5-6679 


5a 


6 a' 


1-97407 


1-162144 


58-8 


3-9481 


6-2724 


6a 


la' 


2-10005 


1-300593 


61-9 


4-2001 


6-8012 


la 


8a' 


2-21297 


1-422097 


64-2 


4-4259 


7-2701 


8a 


9a' 


2-31447 


1-530317 


66-1 


4-6289 


7-6895 


9a 


10a' 


2-40794 


1-627843 


67-6 


4-8159 


8-0716 


10a 


11a' 


2-49320 


1-716587 


68-8 


4-9864 


8-4195 



The first column expresses the radius of the iron conductor in 
terms of that of the other ; the remaining columns are as in the 
first table. 

The expression for the force can by suitably choosing a, a', b 
so that b is somewhat greater than a */2, be made to change sign, 
so that the force between the conductors would then be attractive, 



In the case of two iron conductors 
L = fi +- 2/4, 



1 V M V 

1 °S^' + Xl °S(6'-a«)(&'-a'») 



l <Vlo ^ (62 - a2 - a,2) 

+ -A- 10 a (ft* _„»)(#< _ a '*) 



b 4 (b 2 -a*-a'y I 

+ A lo g(6«_ a *)(6»_ a '«)((6»_. a »)«_ a ^}((6*_ a «)»_ a ^] +etc -J ' 



/here 






In this series the coefficients of the powers of A. rapidly diminish. 
It is perhaps worthy of remark that when /u, is 100 or greater, 
the part depending on the size of the conductors and their 
distance apart is only slightly affected by the value of /x, as A. 
differs but little from unity. 



262 Mr Orr, On the Contact Relations of certain [Nov. 23, 

(2) The Effect of Flaws on the Strength of Materials. By J. 
Larmor, M.A., St John's College. 

The effect of an air-bubble of spherical or cylindrical form in 
increasing the strains in its neighbourhood was examined ; and it 
was suggested that the results might be of practical service in 
drawing general conclusions as to the influence of local relaxations 
of stiffness of other kinds. In particular, it is shewn by the aid 
of the hydrodynamical form of St Venant's analysis, that a cavity 
of the form of a narrow circular cylinder, lying parallel to the 
axis of a shaft under torsion, will double the shear at a certain 
point of its circumference; and the effect of a spherical cavity 
will not usually be very different. For a cylindrical cavity of 
elliptic section, the shear may be increased in the ratio of the 
sum of its two axes to the smaller of them, this ratio becoming 
infinite in accordance with known theoretical principles for the 
case of a narrow slit. It is assumed in the analysis that the 
distance of the cavity from the surface of the shaft is considerable 
compared with its diameter, so that the influence of that boundary 
may be left out of account in an approximate solution 1 . 

The results will however also give the effect of a groove of 
semicircular or semi-elliptic section, running down the surface of 
the shaft, provided the curvature of the surface is small compared 
with the curvature of the groove. 

(3) The Contact Relations of certain Systems of Circles and 
Conies. By W. M C F. Orr, B.A., St John's College. 

(Abstract.) 

The following theorem is first established: — If four circles 
X, Y, P, Q, in a plane or on a sphere, are such that a circle can 
be drawn through one of each pair of intersections of X with P, 
X with Q, Y with P, and Y with Q respectively, (and therefore 
another circle through the remaining four intersections of the 
same circles,) the eight circles which touch X, Y and P, and the 
eight which touch X, Y and Q, can be arranged in sixteen groups 
of four circles, each group consisting of two touching X, Y and P, 
and two touching X, Y and Q, such that each group has two 
common tangential circles besides X and Y. 

A similar result is of course true for groups of circles touching 
P, Q and X, and P, Q and Y respectively. 

The above relation of condition is triply satisfied by the four 
circles that form any Hart-group of circles touching three others 
(restricting the title Hart-group to the eight groups that are 
analogous to the inscribed and escribed circles of a triangle). 

1 See Phil. Mag., Jan. 1892. 



1891.] Systems of Circles and Conies. 263 

Hence, taking as a particular case the inscribed and escribed 
circles of a plane or spherical triangle, the following result is 
obtained : — If we describe circles touching three by three the 
inscribed and escribed circles of a plane or spherical triangle, we 
obtain four sets of four circles, exclusive of the sides of the 
original triangle and its Hart-circle ; each set forms a Hart-group 
and in addition we can obtain twenty -four groups of four circles by 
taking two out of any one set and two out of any other such that 
each group is touched by two circles besides the two circles they 
have been constructed to touch in common. 

Any group whatever of four circles of the eight that touch 
any three given circles satisfy the above relation of condition, 
some singly, some doubly, and some triply; and by taking all 
such groups of four, and describing circles touching them three 
by three the following result is obtained: — Eight circles can be 
described to touch three given circles; these eight form fifty-six 
groups of three ; to touch any three we can describe a set of four 
circles exclusive of the original three, and one which with them 
forms a group of four circles touching four others ; and we can 
form a thousand and eight groups of four circles, two out of one 
set and two out of another, such that each group is touched by 
two common circles besides the two they have been constructed 
to touch in common. 

These theorems are then extended to co-vertical cones which 
are either circular or have double contact with a given one, and by 
projection to conies having double contact with a given one. 

In the last case one of the results obtained is: — If four 
straight lines X, Y, P, Q are such that through the intersections 
of X with P, X with Q, Y with P and Y with Q there can be 
described a conic having double contact with a given one (/>, then 
the four conies touching X, Y and P and the four touching X, Y 
and Q, all having double contact with 0, can be arranged in eight 
groups each consisting of two touching X, Y and P, and two 
touching X, Y and Q, such that each group, besides touching X 
and Y, touch in common two tangent conies having double contact 
with <f> ; and similarly for conies touching P, Q and X, and P, Q 
and Y respectively. 

The enunciation of the reciprocal theorem is obvious. 

Another result is as follows : — Four conies can be described 
having double contact with a given one <j>, and touching three 
given lines or passing through three given points ; to touch any 
three of these four conies sixteen conies can be described having 
double contact with <£, exclusive of the original lines or points 
and four Hart-conics; there are thus four sets of sixteen conies. In 
addition to the groups of four touched by four conies having double 
contact with <f> that can be formed by taking four out of the 



264 



Mr Sharpe, On Liquid Jets under Gravity. [Nov. 23, 



same set, there are thirty-two groups of four conies formed by 
taking two out of any one set and two out of any other, such 
that each group is touched by two conies having double contact 
with the given one </>, besides the two they have been constructed 
to touch in common, and as the four sets can be taken in pairs in 
six ways, there are thus one hundred and ninety-two such groups 
in all. 

Some other theorems are obtained of a more complicated 
character. 

The method of proof is purely geometrical and the first pro- 
position, though not proved as shortly as it might have been, is 
made to depend mainly on a property of Bicircular Quartics 
given by Mr C. M. Jessop in the Quarterly Journal of Mathematics, 
Vol. XXIII., which is equally true for Sphero-Quartics, and which 
for the case of two circles may be enunciated a little differently 
as follows : — If X, Y are any two circles of the same family 
touching two given circles A and B, in a plane or on a sphere, 
and P, Q are any two circles of the other family touching A and 
B, then a circle can be drawn through one of each pair of the 
intersections of X with P, X with Q, Y with P, and Y with Q, and 
of course another circle through the remaining four intersections 
of the same pairs. 

(4) On Liquid Jets under Gravity. By H. J. Sharpe, M.A., 
St John's College. 

1. The motion (Fig. 1), which is in two dimensions, is sup- 
posed to be symmetrical with regard to x'Ox which is the axis 
of the vessel and jet. BEF is the semi-outline of the vessel, 
FJ of the jet. AF is the semi-orifice, which is small compared 
with the dimensions of the vessel and the depth OA of the 
liquid. Gravity acts parallel to x'Ox. OE is the surface of the 







y 




r 


K 








~~1e 








B 








Fig.i- 




g 


\ 


F 


J 



liquid, which is maintained steady. AF is supposed to be so 
small that it may be considered either as the arc of a circle 



1891.] Mr Sharpe, On Liquid Jets under Gravity. 



265 



with centre in the surface of the liquid, or as a small straight 
line perpendicular to Ox. For simplicity we shall take OA the 
radius of the circle (or the depth of the liquid) as unity. 

If g be the acceleration of gravity referred to this unit it 
will be convenient to put 

a 2 for 2g (1). 

We shall take as the origin of Cartesian and Polar Coordinates 
x, y, r, 6 and we shall put 

x for 0-1) (2). 

Let % be the stream function to the right of AF ; u y v the 
velocities parallel to Ox m Oy. 

Further let AF=ir/p (3), 

where p is a large number. 

On the right of AF we will take 



JL = u = ar l cos £0 + % Cn ' e P nx ' cos pny 



dy 
ax 



CO 



•ismifl + SoV*"*' 



sm pny 



.(4), 



where c n ' is an arbitrary constant. X indicates summation with 
regard to n for all integral values from 1 to oo . 

Therefore along AF, we have on the right of it at every point, 
nearly 

u = a + $c n ' cos pny ) ^ 

v = — \ay + 2c„' sin pny) 

[Of course AF is really half the small segment of a circle. 
The equations (4) and (5) are only approximate (the more so 
the larger p be taken) but it will be pointed out afterwards 
(Art. 5) how their exact forms can be found — forms which would 
be suitable for all values of p — but as these are rather complicated 
it is better to begin with the simplest case first.] 

Let y{r be the stream function on the left of AF, and 
on the left of AF we will take 






cos my) + %c n € pnx ' cos pny + A 



— -r-= v = — S {a^*^ sin my) — %c n e pnxf sin pny 



•(6), 



where a m c n and A are arbitrary constants, and S indicates 
summation with regard to m for a finite number of values of 
m, the largest of which is supposed to be small compared with p. 

VOL. VII. pt. v. 21 



266 Mr Sharpe, On Liquid Jets under Gravity. [Nov. 23, 

Therefore along AF we have on the left of it at every point 
11 = S (a m cos my) + Xc n cos pny + A } . ^ 

v = - 8 (a m sin my) — %c n sin pny ) 

But as the motion must be continuous through AF, the w's 
and v'a in (5) and (7) must be the same. Therefore we get 
8 (a m cos my) =a-A+% (c,/ - c n ) cos pny) ^ 
- S (a M sin my) + ±ay = % (c.' + e.) am pny J 

These must hold from y = to y = ir/p. But if we expand 
the left-hand sides of (8) by Fourier's Theorem, we can get c n and 
c ' as functions of n. 

n 

2. Doing so, we shall get 

c +c=2S/a sin — , „ \ ....(9). 

" n J ra j> 2 2 mV J p « 

\ W7r """7"/ 
c'-c — aSf/n .^sin-^. C ° SW7r a A (10). 

Also ji-a + iSffa,, x -^- sin — ) = (11). 

Also in order that the second of the equations (8) may hold 
at the limit y = wjp, we must have 

-S^,^— ) + 2 - = (12). 

We shall now prove an important property of c„ and c„\ 

As 1 is the least value of n, and as by hypothesis m/p is always 
a small fraction, we may always (if need be) safely expand the 
fractions in (9) and (10) in ascending powers of m'/pW. 

Doing so in (9), we shall get 

mir cos nir I ' m 2, „ \) a cos mr 

— . l + -2-2 + &c. n- — . 

p wrr \ pn J) p n 

(13). 

But by (12) the first and last term here disappear, so that 
(c^' + cj is always a small quantity at most of the order 1/p 3 . 
Similarly from (10) we see that (c n ' — c n ) and therefore c n and c„' 
are always small quantities at most of the order 1/p 2 . We say 
' always ' — even when n = 1 when they are largest. 



C ;+c„ = 2,Sf ja>n- 



1891.] Mr Sharpe, On Liquid Jets under Gravity. 2G7 

3. From (G) the equation to BEF is 

S fa e mx ' sin my) + 2 ^ e pii * sin F?i y + /ty 

= £ -^sin ■ — ) + (14). 

\m p ) p 

If y = b is the equation to the asymptote to BEF 

ji a (Q> m ■ ??i77 "\ Air ,, _. 

J.6 = £ ^- ra sin — + — (15), 

so that if 6 be finite, .4 is a small quantity of the order 1/^. 

Looking now at (6) we see that if OE is to be the surface 
of the liquid, u and v must, when x' = —1, be small quantities 
at most of the order ljp. A and the X term already satisfy 
that condition. In the S term m has several values, enough 
to satisfy conditions (11), (12) and (15). Suppose the particular 
m in (6) to be the smallest of these values, and suppose m = \ogp, 
then when a/ = — 1, the 8 term also satisfies the surface condition, 
and the more accurately the larger p is, since \ogp/p diminishes, 
as p increases. 

If FJ is to be a jet we must have, since AF is small, at every 
point of the jet, nearly 

w 2 + v* = 2gr. 

But we see at once from (1) and (4) and Art. 2 that this con- 
dition is fulfilled, the error being of the order 1/p 2 . 

4. To get some idea of the maximum value of this error, we 
see from (4), since at F we have nearly 

u = a — c/ 
that — 2c//a is a fair measure of this maximum. 
From (10) and (13) 

c; = l ^S{m*aJ (16). 

From (11) and (12) we have nearly, since A is a small 
quantity, 

-a + S(aj = 0, and ^-S(ma m ) = 0. 

If for instance we take 8 and 9 for the two values of w, then 
p will be about 2981 and the maximum error about 

+ -0000143. 

21 2 



268 



Mr Sharpe, On Liquid Jets tinder Gravity. [Nov. 23, 



If we took a sufficient number of values of m to satisfy, in 
addition to previous conditions, the condition S (w 2 a m ) = 0, we 
see from (16) that the maximum error would be of the 3rd order 
of smallness, and so on for higher orders. 

5. Suppose p instead of being large, were somewhat smaller, 
we should then proceed thus. 



'■ F 



Fig. 2 



NA 



From F (fig. 2) draw FN perpendicular to Ax. 
Let NF = irjp' where 



7T . ir 
—: = sin — , 
p p 



.(17). 



In equations (4) to (8) &c. put p' for p, and consider (4) to 
apply to the right, and (6) to the left of NF. (8) also would have 
to hold from y = to y = ir\p'. Of course from (17) p' could be 
expressed as accurately as desired in terms of p. 

6. From (4) the equation to the outline FJ of the jet is 



Zd $ . o a ^ C n —pnx' • 

— r^sm^e + X— e * smpny- 



2a . 3tt 
_ sm¥ ....(18). 



As the X term is of the order 1/p 3 , we see that in all solutions 
obtained by the present method the shape of the jet is nearly 
independent of the shape of the vessel, and is dependent only 
on the angle which the orifice subtends at 0. 

Upon this point light may be thrown by the following 
Article. 

7. Since writing the preceding, I have examined somewhat 
carefully equation (14) which gives the outline of the vessel — 
in the case where m has the values 8 and 9. In this case a m 
and a n are determinate. I find that in (15) b is not perfectly 
arbitrary, but appears to have limits in order that the curve 
BEF may be continuous. I have tried to take it as large as 
possible. I have actually taken it = 2tt/9 or about -6981, but 



1891.] Mr Sharpe, On Liquid Jets under Gravity. 



269 



whether this is the largest admissible value of b (for ra = 8 and 9) 
I am not sure. The result is that I get a curve something of 
this shape (fig. 3) for BEF. 



Fiff. 3 • 




One noticeable feature is the existence of a long spout or pipe- 
like portion OF before we reach the orifice F, which may perhaps 
explain the result noticed in Art. 6. 

I am afraid this spout exists in all solutions obtained by the 
present method. There is an interesting point about the curvature 
of the outer stream-line in the neighbourhood of F. It will be 
found from equations (14) and (18) that there is a sudden change 
of curvature at F — that the curvature on the right of F divided 
by curvature on left of F gives a small quantity of the order 
1/p, thus corroborating (as far as the approximate character of 
the present method will allow) a remark of Kirchhoff found at the 
end of Art. 96 of Lamb's Motion of Fluids. 

(5) Theory of Contact- and Thermo- Electricity. By J. 
Parker, M.A., St John's College. 

In this paper, which treats of an electrified system of metals 

situated at rest in a vacuum in an unvarying state, we shall use the 

electromagnetic C.G.S. units, so that if Q, Q' be the charges of 

two small bodies at a distance of r centimetres, the electric 

QQ' 
repulsion between them is e -^- dynes, where e is the constant 

87 x 10 19 . 

We first require to know the energy U and entropy (f> of our 
system. The system being necessarily so chosen that its total 



270 Mr Parker, On Contact- and Thermo- Electricity. [Nov. 23, 

charge is zero, let us suppose that by altering the relative positions 
of its parts, every one of its charges becomes zero, and denote the 
corresponding values of the energy and entropy by U and <£ . If 
from any cause, such as the parts of the system not being sufficiently 
numerous, this (or any other) operation cannot be directly per- 
formed, we may always make use of subsidiary bodies; and the 
final result being independent of the subsidiary bodies employed, 
we may argue as if they were entirely unnecessary. 

For the energy U, we assume Helmholtz' expression 

U=U +e$ < ^ + Q A F A + Q«F B + 



= U +6{iQ A V A + %Q s V B + } + Q A F A +Q B F B + (1), 

where A,B,C, ... are the different homogeneous bodies of uniform 
temperature which the system contains; Q A , Q B , Q c , ... their 
charges; V A , V B , V c , ... their potentials; and F A , F B , F c , ... 
quantities which depend respectively on these bodies but not on 
their electric states. In what follows, the states of the different 
bodies will be completely defined by their temperatures, so that 
F will be a function of the temperature depending in form on the 
nature of the metal. 

To obtain the value of <£, we make use of Joule's law that if a 
steady current I be flowing in a homogeneous body of uniform 
temperature and resistance R, the heat evolved is RP ergs per 
second. From this we deduce that while a quantity Q is being 
transmitted, the heat evolved is RIQ ; and therefore, since R is 
independent of /, as / diminishes, the heat evolved when a given 
finite quantity of electricity is transferred, diminishes in the same 
ratio as /. Now the process becomes more and more nearly 
reversible as / diminishes, but does not actually become reversible 
until / vanishes. Hence if our given system be made to undergo 
a reversible operation of any kind in which no part of it is com- 
pressed or distorted, and no charge made to pass from one body to 
a different body or to a body of the same kind but at a different 
temperature, there will be no thermal effect produced and conse- 
quently no change of entropy. So long therefore as the charges 
are not made to leave the bodies on which they were at first, the 
entropy of the system is unaltered by any change in the distribution 
of the charges or in the relative positions of the bodies. We may 
therefore put 

4> = ir A (Q A ) + ylr B (Q B ) + f c (Q c )+ , 

where ^r A (Q A ) only depends on the body A and its charge, ^jr B (Q £ ) 
only on B and its charge, ...... 



1891.] Mr Parker, On Contact- and Thermo- Electricity. 271 

Now take a second system identical with the first, forming 
with it a compound system whose entropy is 2(f>. By a reversible 
process, such as we have already described, let a charge be made to 
pass from one metallic body A to the other body A, and suppose 
that no other charge passes from one body to another. Then, since 
by what precedes the entropy of the compound system is unchanged, 
it follows that, if q be the final charge of one body A and 2Q A — q 
of the other, yjr A (q) + ^ A (2Q A — q) is independent of q, whatever q 
may be. We therefore infer that ty A {Q A ) = ^(0) + Q A H A , where 
H A is independent of Q A . 

Next, let us take a system formed of the original system <p and 
of a second system which also contains a metal body A at the same 
temperature as the body A in the first system but different in 
form and size and with any chai'ge Q A . Let H A be the quantity 
corresponding to H A . Then, by supposing any charge to pass from 
one metal A to the other metal A, without the passage of a charge 
between any other bodies, we find H A = H A . Hence H A is inde- 
pendent of the size and form of A as well as of its electric state. 

Thus finally 

<f> = f A (0) + ^ B (0)+...+Q A H A + Q B H B +... 

= <f>o + Q^ A + Q B H B + (2). 

To complete the expressions for IT and $, we require an im- 
portant identical relation which holds between F and H. Let the 
metal A and any other part which is at the same temperature 6 A 
be slowly heated to 6 A + dd A , and let the parts of the system be at 
the same time slowly moved about so that no charge passes from 
one body to another. Then we have 

dU=dU +edX^- + Q A dF A +...) 

dcj> = d<f> + Q A dH A + ... J 

If dW be the work done on the system, dll — dW is the heat 
absorbed, and since the operation is reversible, we have 

dU-dW=e A d<j>, 
or dU -6 A dcf>- dW+ ed?,QQ-+ Q A dF A + ... 

-QJJH A -Q B A dH B -...=O. 

Now take a second system identical with the first except that 
every charge is reversed in sign, and let it undergo a reversible 



272 Mr Parker, On Contact- and Thermo-Electricity. [Nov. 23, 

operation similar to that just described. Then since dU , d<f> , dW, 

00' 
and edZ -^- are the same as before, we obtain 
r 

dU-d A d<f>-dW+ edZ^ - Q A dF A - ... 

+ Q A A dH A +Q B A dH B + ., : =O. 

These two relations being true for all values of Q A , Q B , ..., it 
follows that 



dF\ 
d6 A 



dJL 

dd. ' 



.(3). 



To obtain the theory of the Peltier effect and the correspond- 
ing change of potential, let two long wires A, B, of different metals, 
be joined together at J, and also connected, as in the figure, with 



Iron 



B 3 & 



Iron 



two plates A, B, which are respectively of the same metals as the 
two wires. Parallel and opposite to these two plates place equal 
plates of any the same metal, as iron, and connect the iron plates 
by long iron wires with each other, or with a large distant mass of 
iron in the neutral state, so that the two iron plates are always at 
potential zero. Also, to make the calculations simple, let us 
suppose the air exhausted about the plates ; but, to make the 
results general, the junction J must be surrounded by air, and to 
prevent the air coming near the plates, it must be enclosed in a 
bag and the junction J and the bag kept at a great distance from 
the plates. Then when the system is at a uniform temperature 0, 
if Q B , Q A be the charges of the plates ; V B , V A the potentials ; U 
and <fi the energy and entropy of the system, we have 



•(4), 



U=U + e [IQ B V B + iQ A V A ] + Q B F B (6) + Q A F A {6) 
^ = <f> o + Q B H B (0) + Q A H A (e) 
where U and </> will remain constant in what follows. 

If now by slowly moving the plates B nearer together and slowly 
separating the plates A, any quantity of electricity q be made to 
pass slowly from A to B against the abrupt rise of potential V B — V A 
at J, and the temperature of the system be kept constantly equal 



1891.] Mr Parker, On Contact- and Thermo-Electricity. 273 

to 0, there will be no thermal phenomenon in the system except at 
J, and the heat absorbed there will be times the increase of 
entropy, or q0(H B - H A ). This may be written qP BA , and we see 
that Jr BA = Jr AB , Jr CA = Jr CB -f- Jr BA - 

Again, the work done on the plates B is — \qeV B , and the work 
done on the plates A, \qsV A ; so that the total work done on the 
system is -%qe (V B - V A ), or - \qB BA , if D BA stand for e(V B - V A ), 
or the electromotive force of contact. Hence, since the increase of 
energy is \qD BA + q(F B - ^a)> we nave 

%D BA + F B -F A = --hD BA + P BA , 
or D BA + F B -F A = P BA =6(H B -H A ) (5). 

Combining equation (5) with the identity -^ = -^ , we get 



d^BA JT TT 

dd ~ ' A ' 

d (D BA \ F B -F 
dO\e~) 2 

de\ e )~ dd dd 0d6 y 

dDi 

d0 



y (6), 



A _ dHg _ dH^ _ 1 t\ F _ „ , 
)~ dd d0 ~0d0 K B Ah 

and P„,= 



dD 
The result P=0 -=%■ has been given on four independent 
dv 

occasions : — by Prof. J. J. Thomson in his Applications of Dynamics 

to Chemistry and Physics ; by Maxwell in his small Treatise on 

Electricity, where he has abandoned the older assumption that 

P = D; by Duhem ; and, lastly, by the present writer. 

Next, let the plates A and B be of the same metal in the same 
molecular state but at slightly different temperatures 0,0 + d0. 
Then if V and V be potentials of A and B, Q and Q' their charges, 
we have 

TT=U o + e{hQ'V' + lQV} + Q'F(0 + d0) + QF(0), 

cj> = cf> o + Q'H(0+d0) + QH(0). 

If therefore we suppose the change of temperature at J to be 
so gradual that the heat conducted across the junction while a 
small charge q passes slowly from one plate to the other, can be 
neglected, we easily find, if X be the ' specific heat of electricity,' 
that is, the coefficient of the Thomson effect, or %d0 the quantity 



274 Mr Parker, On Contact- and Titer mo- Electricity. [Nov. 23, 
which corresponds to P BA of equation (5), 

€ (V'-v) + ^dd=td0=e~de. 

do do 

Hence t " e W"fe- | (7). 

and therefore V — V = J 

The result F' — V = 0, which asserts that there is wo electro- 
motive force of contact between two pieces of the same metal at 
different temperatures, is of the utmost importance. At first 
sight it may seem to be in contradiction to experiment ; but on 
closer examination, as we shall shew later on, this is found not to 
be the case. 

Equations (5), (6), and (7) contain the whole theory of the 
Peltier and Thomson effects. They enable us to discuss the 
properties of themoelectric circuits, and the results thus obtained 
include all those of Thomson and others which have been tested by 
experiment. After shewing this, we will point out the close 
analogy between our theory of thermoelectric circuits and Helm- 
holtz' theory of the galvanic battery. Then we will consider some 
experiments bearing on our theory. 

In the first place, we obtain, from equations (6) and (7), 
Thomson's result 

d fJr Bi \ 2< B — z< A /gv. 



d6 

Next, let two pieces A, A', of the same kind of metal, be con- 
nected by a piece B of a different kind of metal, and let the free 
ends of A and A' be at the same temperature 6, while the junctions 



a, 



A B A' 

of B with A and A' are at the different temperatures 6 X , 2 , re- 
spectively. Then since there is no electromotive force of contact 
between two pieces of the same metal at different temperatures, it 
follows that, in the state of equilibrium, the potential of the free 
end of A' will exceed that of the free end of A by 

This will not generally be zero, and therefore, as experiment also 
shews, if a circuit be formed by joining the free ends of A and A', 
equilibrium will be impossible and there will be a current, called 
a thermoelectric current. 



1891.] Mr Parker, On Contact- and Thermo- Electricity. 275 

If A be a piece of metal in the same molecular state through- 
out whose temperature varies in any gradual way we please from 
end to end but has the same value at both ends, the ends will be at 
the same potential, and therefore if they be joined so as to form a 
circuit, there will be no current produced. This is the result 
obtained experimentally by Magnus, who found it impossible to 
obtain a current by unequal heating in a homogeneous circuit. If, 
however, we take a homogeneous circuit, and by filing make a 
junction of a very thick piece and a very thin piece, it was found 
by Maxwell that on applying a flame to this junction, a current 
is produced. 

Now let a thermoelectric circuit be formed of two different 
metals A, B, as in the figure, and let the temperature of every 

e^> >- B 



A 

part of the circuit be kept constant ; and O being the tempera- 
tures of the junctions. Then the 'electromotive force of the 
circuit' is defined to be e times the sum of the abrupt rises of 
potential as we travel round the circuit in the direction of the 
current. Hence since the electromotive force of contact of two 
pieces of the same metal at different temperatures is zero, if the 
current be supposed to flow from A to B through the junction of 
temperature 6, the electromotive force E will be given by 

E = D-D, (9). 

For a circuit formed of several metals, we shall have 

E = XD (9)'. 

This result, which has not been tested directly by experiment, 
has been given by Duhem and assumed by Clausius. 

When the current is steady, let I be its strength and R the 
resistance of the circuit. Then 

E = RI, 

since the sum of the abrupt rises of potential at the various 
junctions must be exactly balanced by the gradual fall of potential 
in the other parts of the circuit. 

Confining ourselves, for the sake of simplicity, to the case of 
a circuit formed of only two metals, as in the preceding figure, 
the heat absorbed in a second at the junctions 0, O , will be 

(P - P ) /h I [{D + F B (0) - F A (0)} - [D + F B {0 O ) - F A (0 O )}] 

= 1{E+ {F B (0) - F B (0 O )} - [F A (0) - F A (0 O )}], 



276 Mr Parker, On Contact- and Thermo-Electricity. [Nov. 23, 

which is exactly equal to the heat evolved in the rest of the 
circuit. 

We can now give the principal results obtained by Sir W. 
Thomson, who avoids entirely the question of the electromotive 
forces of contact at the various junctions, either of two different 
metals, or of two portions of the same metal at different tempera- 
tures. 

Combining equation (9) with the results 



P = 0^ and 



dd\ e )~ 



de dd\ e j e 

we get Thomson's formulae 

E=\ e ^de=\\H B -H A )de^ 

F =C ■ \ do). 

dE _ dP BA ,~ ^ . 
dd~~aW ^-^ 

From the second of these equations, we see that if the cur- 
rent tends to cool any junction of two metals in passing through 
it, the electromotive force will be increased by raising the tem- 
perature of that junction. The electromotive force, as far as it 
depends on that junction, will be a maximum when P vanishes. 
As further increases, P will become negative and the electro- 
motive force will diminish. The temperature T at which P 
vanishes is called the ' neutral point ' of the two corresponding 
metals, or of the circuit when it is composed of these two metals 
only. 

Again, when and O are so nearly equal that we may put 
6 Q = — t, where t is small, we have 

dD r 2 d 2 D 
T dd + 2d6» '"' 

and therefore for a circuit of two metals 

P r 2 d fP\ 



„ P t" d /P\ 

E = T J-Jdd[w) + 



When the 6 junction is at the neutral point T, this gives Thom- 
son's formula 

* = "!?§ ("> 



1891.] Mr Parker, On Contact- and Thermo- Electricity. 277 

The current is then of the second order only and flows through 
the circuit in the same direction whether the other junction be 
hotter or colder than T. 

When P does not vanish, we have another of Thomson's 
formulae 

E=r^ (12). 

For simplicity taking t to be positive, we see that E and P are 
of the same sign. The current therefore travels in such a 
direction as to absorb heat at the hotter junction of the two 
metals. 

When the temperatures of the two junctions differ by a finite 
amount, it will be seen from (9) that the electromotive force is 
generally finite even when one junction is at the neutral point. 
If we suppose that the hot junction is at the neutral point, it is 
clear that both the Thomson effects cannot be absent, for then 
the heat that is developed in the homogeneous parts of the 
circuit would be all conveyed by the current, without assistance, 
from the coldest part of the circuit. It was the consideration of 
this case that led Sir W. Thomson to the discovery of the 
' specific heat of electricity.' 

If we had accepted the old assumptions that the thermal 
effects measure the electromotive forces of contact, or that P = D 
and that the electromotive force of contact of two pieces of the 
same metal at slightly different temperatures is %dd, we should 
have had 



E=p-p -(\i B -t A )de (is). 

J n 



Now this is the very equation that is obtained by writing down 
the condition that the sum of the quantities of heat absorbed at 
the four junctions is equal to the heat evolved (according to 
Joule's law) in the rest of the circuit. Here then the two as- 
sumptions that P — D and that the electromotive force of contact 
of two pieces of the same metal is XdO, exactly neutralize each 
other ; but this fact, it is clear, does not prove that the assumptions 
are correct. 

It may be noticed that the assumption P = D cannot agree 

dD 

with our result P = 6 -^ unless P is proportional to 6, which 

experiment shews is not usually the case. Again, if X B = 2^ = 0, 
equations (7) shew that F B , H B , F A , H A are constant, and there- 
fore, by equation (6), P — D= Cd, where C is independent of 6. 
But if P=D =Cd, it does not follow conversely that F B , H B , F A , H A 
are constant, nor that % B = 2^ = 0. 



278 Mr Parker, On Contact- and Thermo- Electricity. [Nov. 23, 

Returning to our own theory, let us take as a first approximation 
D = a + b0+c0\ 
where a, b, c are constants. Then 

P = 0(b+2c0). 
But P vanishes when = T : hence 

b + 2cT=0. 

Thus we have 

E=D-D o = b(d-0 o ) + c(0 2 -e*) 

= -2cT(0-0 o ) + c(6*-0*) 



= -2c(0-0 o )J2V^| (14), 



the formula of Avenarius and Tait, which has been found to agree 
sufficiently well with experiment. 

... d (D\ F B -F A 

Again, since S^J = ~ ^~ > 

we find F B -F A =-a + c0\ 

which is satisfied by taking 

F=k0* + F, 

where k and k' are constants. 

dP 
Hence -^ = 2kd, or the ' specific heat of electricity,' %, varies 

as the absolute temperature. Also -^ — 2k, or H = 2k0 + I. 

do 

The ' thermoelectric diagram ' of Prof. Tait is greatty simplified 
by our results. For if co be the standard metal and M any 
other metal, the ' thermoelectric power ' of M with reference to co, 

or \ a" , is equal to H M — H m . The standard metal is taken to 
do 

be lead, because the ' specific heat of electricity ' of lead is zero, 

or H M constant. Hence in the diagram, the abscissa represents 6', 

and the ordinate, Hm — H*, where H M is constant. 

Lastly, let a galvanic battery have both poles of the same 
metal, and let every part of it be kept at the constant tempe- 
rature 0. Let L be the heat evolved when we effect in any 
way at constant pressure the same chemical change as is produced 
by the passage through the battery of unit quantity of electricity 
in the direction in which the battery tends to give a current. 



1891.] Mr Parker, On Contact- and Thermo- Electricity. 279 

Also let us suppose that when the battery is in action, the 
chemical changes which take place are reversible, and the Peltier 
effects and electromotive force the same as when the current 
is infinitesimal. Then the heat absorbed at the junctions on 
the passage of unit quantity can be easily proved to be 

a dE(. p a dD\ 

6 W (just as P = M ). 

Hence since the heat evolved in the homogeneous parts in the 
same time is E, and the total heat evolved L, we have the result 
of Helmholtz and Gibbs, 

L = E - e % < 15 >- 

It now only remains to look at the experimental evidence 
relating to the contact theory. In so doing we must remember 
that an electromotive force of contact is produced not only by 
the contact of two conductors, but by the contact of a conductor 
and a non-conductor, and also, but less easily, by the contact of 
two non-conductors. Again, for the sake of simplicity we shall 
often follow the custom of writing B\ A for D BA , and whenever 
it is necessary to indicate the temperature, we shall use suffixes : 
thus B/A e denotes that the two different bodies A, B, are at 
the same temperature 6; B e jB eo , that two portions of the same 
substance are at the different temperatures 6, 6 . 

It has been pointed out by Maxwell that in experiments 
like those of Clifton and Pellat, in which two plates Z, C, of 
different metals, are employed in the open air, we really measure 
the sum A/Z+Z/C+ C/A, or D zc ,+ A/Z -A/C. In like manner, 
when we employ two plates of the same metal Z but at different 
temperatures 6, 0> we measure Z e /Z 9o + A/Z e — A/Z 9o . Now 
hitherto the terms A/Z 9 — A/Z 9o have been alwaj^s omitted, and 
the experiment has been supposed to prove that Z 9 /Z 9a is not 
zero. But clearly we cannot assume that 

A/Z O -A/Z 0O = 0, or that A/Z 

is independent of the temperature ; and therefore the experiment 
does not contradict our result that Z 9 /Z 9o = 0. 

Let us assume that the thermal effects measure the electro- 
motive forces of contact ; and suppose that we repeat the ex- 
periments of Clifton or Pellat with two plates Z, G, of different 
metals, first at the temperature 6, and then at a slightly different 
temperature. From the results obtained, we find the value of 



280 Mr Parker, On Contact- and Thermo- Electricity. [Nov. 23, 

which we will call f x {&). Next, if two plates of the same metal Z 

be employed but at slightly different temperatures, we obtain 

d 
X z + -jn {AjZ) = a known quantity f 2 {&). Similarly we may get 

We then easily find 

^f - (Z z - X c ) =f x {ff) - {/, (0) -f 3 (6)}, 
a result in which the influence of the air does not appear. 

If we accept the thermo-dynamical theory here developed, 
these experiments give 

^ + d_ {A/z _ A/c}=fi(0)> 

±(A/Z)=f t (0), ^(AIG)=f 3 (0), 
and therefore 

/, W - I A W - /. OT) - ^jf = l f 

which is the very result we obtained by assuming that the thermal 
effects measure the electromotive forces of contact. Hence experi- 
ments like those of Clifton and Pellat do not help us to decide 
which of the two theories is correct. 

Again, on the assumption that P = D, it follows from the 
experiment which gives D zo + A/Z — AJG, since P is very small, 
that the electromotive forces of contact between the plates and 
the air form great part of the phenomenon observed. It might 
be supposed that this conclusion could be tested directly by ex- 
periment by exhausting the air; but Pellat, by reducing the 
pressure to 1 cm. or 2 cms. of mercury, found little difference in 
the result. 

If we take the thermo-dynamical theory, it will follow that 
in Pellat's experiment with two plates of the same metal at 
different temperatures, the whole of the phenomenon observed is 
due to the contact of the plates and air. In the case of two 
plates of different metals at the same temperature, a large part 
of the phenomenon observed will also probably be due to the 
contact of the metals with the air. Taking into account the 



1891.] Mr Parker, On Contact- and Thermo-Electricity. 281 

experiments of Brown and Pellat on the effects of reducing the 
pressure of the air, we conclude that within attainable limits of 
pressure, AjZ, the electromotive force of contact of the metal 
and air is nearly independent of the pressure of the air. It will 
now be shewn directly from theory not to be unreasonable to 
suppose that this may be the case. 

If, for shortness, we take AjZ to be a volt, or 10 8 times the 
electromagnetic C. G. S. unit, and suppose the distance, d, of the 

two electric layers to be „ .,,-, cm., the attraction between the 
J 3 x 10 

metal and the air per square centimetre will be 

1 (A/Z\* 

or about 40,000 x 10 6 dynes. Hence since the pressure of one 
atmo is only about 10 6 dynes per square centimetre, we can partly 
understand why the reduction of the pressure to 1 cm. or 2 cms. of 
mercury produces so little effect. 



VOL. VII. pt. v. 22 



COUNCIL FOR 1891—92. 

President. 
G, H. Darwin, M.A., F.R.S., Plumian Professor. 

Vice-Presidents. 

T. M C K. Hughes, M.A., F.R.S., Professor of Geology. 

J. J. Thomson, M.A., F.R.S., Professor of Experimental Physics. 

John Willis Clark, M.A., F.S.A., Trinity College. 

Treasurer. 
R. T. Glazebrook, M.A., F.R.S., Trinity College. 

Secretaries. 

J. Larmor, M.A., St John's College. 
S. F. Harmer, M.A., King's College. 
E. W. Hobson, M.A., Christ's College. 

Ordinary Members of the Council. 

W. Gardiner, M.A., F.R.S., Clare College. 

W. Bateson, M.A., St John's College. 

A. Cayley, Sc.D., F.R.S., Sadlerian Professor. 

W. J. Lewis, M.A., Professor of Mineralogy. 

W. H. Gaskell, M.D., F.R.S., Trinity Hall. 

A. Hill, M.D., Master of Downing College. 

A. S. Lea, Sc.D., F.R.S., Gonville and Caius College. 

A. Harker, M.A., St John's College. 

L. R. Wilberforce, M.A., Trinity College. 

H. F. Newall, M.A., Trinity College. 

C. T. Heycock, M.A., King's College. 

A. E. H. Love, M.A., St John's College. 



PROCEEDINGS 



OF THE 



Cambrifcrg* ^^iksop^iral Stodeig. 



Monday, Feb. 8, 1892. 
Professor Darwin, President, in the Chair. 

Mr W. Heape, M.A., Trinity College, was elected a Fellow 
of the Society. 

The following communication was made to the Society : 

Long Rotating Circular Cylinders. By C. Chree, M.A., Fellow 
of King's College. 

§ 1. In Vol. vil, Pt IV., pp. 201 — 215 of the Proceedings I found 
a solution for a thin elastic solid disk of isotropic material rotating 
with uniform angular velocity about a perpendicular to its plane 
through its centre. In the present paper the same method is 
applied to a long right circular cylinder of isotropic material 
rotating about its axis. The cross section of the cylinder when 
solid is supposed of radius a, when hollow its outer and inner 
boundaries are of radii a and a' respectively. The axis of the 
cylinder is taken for axis of z, the origin being at the middle 
point, and the notation for the displacements, strains, stresses, etc. 
is the same as in my previous paper, except that the dilatation is 
denoted by A. 

If Poisson's ratio, v, be zero the solution obtained here satisfies 
all the internal and all the surface equations, whatever be the 
ratio of the length, 21, of the cylinder to its diameter. But for 
other values of n it is only true to the same degree of approxi- 
mation as Saint- Venant's solution for beams, and like that solution 
can legitimately be applied only when l/a is large. This re- 
striction of Saint-Venant's solution, whether for torsion or flexure, 
is not perhaps in general sufficiently recognised, but the best 
authorities I believe regard it as sufficiently exact only when the 

VOL. VII. PT. VI. 23 



284 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8, 

length of the beam attains to something like ten times its greatest 
diameter. Unless 77 = the present solution ought to be simi- 
larly restricted, and it should not be applied to the portions of the 
cylinder immediately adjacent to its ends. The solution con- 
siders solely the action of the "centrifugal force", taking no 
account of gravity or of the action of any forces applied at the 
ends by the bearings. 

§ 2. The solution satisfies exactly the internal equations — 
viz. (2) and (3) I.e. p. 203 — , and also the conditions at the 
curved surface or surfaces. The only condition it fails to satisfy 
exactly is that Hz vanish at every point of the flat ends. In- 
stead of this we have to be content with satisfying 



J a 



2irrz7dr = 0, 



i.e. instead of making the normal stress over every element zero 
we make the resultant normal stress zero. The solution is thus 
based on the principle of statically equivalent load systems — 
referred to in my previous paper, pp. 206-7 — and so can be 
regarded as satisfactory only when the dimension 2a is small 
compared to the dimension 21. 

The solution is found by starting with the expressions (16), 
p. 205, for the displacements, which satisfy the internal equations. 
We have then to determine the arbitrary constants by means of 
the surface conditions, viz., 

^ = = rr, when r = a and when r = a', 



rz = = I 2irr zz dr when z = ± I. 



§ 3. It is unnecessary to reproduce the algebraical work, as 
the solution may easily be verified. In terms of Young's modulus 
E and Poisson's ratio rj it is as follows : 

2/2 *2\ 3 — 5?; 2 3 (1 - 277) (1 + 77) 

u = <o* P (a» + a>)r HE{l * v) -co* P r^ ^((-1) 

L 2 aV i! ( l+77)(3-2r;) 

w = - a ?p(a* + a'*)z 1 £ B (2), 

A „W. ^ (l-^)(3.-q) . ,(l- 2iy)(l+q ) 

A = a>p(a+a) ^ {1 _ v) -»P» 2E(1- V ) ' m " {6) * 

^ = a>>(a 2 -r 2 )(l-a7r 2 ) 8 -^|^ (4), 



1892.] Mr Ghree, On Long Rotating Circular Cylinders. 285 

♦Wp^ + a' + aVy^^^Xgl^ (5), 

J^VK + .'-^j^L. (6), 

-=0 (7). 

For a solid cylinder the displacements and stresses may be 
correctly deduced from the above by omitting all the terms con- 
taining a' 2 . This solution for a solid cylinder is identical with 
one deduced from the case of a rotating spheroid* by supposing 
the ratio of the axis of figure to the perpendicular axis to increase 
indefinitely. 

§ 4. In what follows I shall assume and "5 as limiting 
values of vj~. As regards the latter limit there is a difficulty that 
will be best understood by reference to the relations 

n/m = 1 — 2v, 3n (1 — n/Sm) — E 

between E, rj and Thomson and Tait's elastic constants. 

If we regard E as finite, we must when n = h, have m 
infinite. Thus those terms in the expressions for the stresses in 
terms of the strains which contain m as a factor may remain 
finite, though the corresponding strains vanish in the limit when 
77 = ^. This explains the apparent inconsistency in the express- 
ions supplied by our solution for this value of 77. The strains in 
this case in the solid cylinder take the remarkably simple forms 

u = co 2 pa 2 r/8E, w = - co 2 pa 2 z/4<E, A = (8) ; 

so that the three principal strains, viz. -5- , u/r and -=- , are every- 
where constant. The vanishing of A is a necessary consequence 
of m being infinite, for this implies that the material is incom- 
pressible. 

§ 5. The expressions for the strains and stresses in the axis 
of a solid cylinder and at the inner surface of a hollow cylinder in 
which a/a is infinitely small are, it will be noticed, totally 
different ; for in the former case terms in « /2 /r 2 simply do not 
exist, whereas in the latter case a' 2 /?' 2 = 1. There is thus, as in 
the thin disk, a discontinuity in passing from a solid to a hollow 
cylinder however small a/a may be. At first sight this appears 
absurd, for it may be argued that if matter has a molecular 
structure, as is generally supposed, then cavities exist everywhere 
between the molecules, and there is no reason why a cylinder 



* Quarterly Journal..., vol. xxiii., 1889, p. 23, Equation (103). 
t Phil. Mag. September 1891, pp. 235—6. 



23—2 



286 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8, 

apparently solid should not have a cavity or cavities devoid of 
molecules occupying the whole or a great portion of its axial 
length. Would not then, it might be urged, such a solid cylinder 
act according to our solution quite differently from another of the 
same material in whose axial line there happened to be numerous 
molecules ? The following seems a satisfactory explanation of this 
difficulty : 

A hollow cylinder in the mathematical sense is one in which 
7z and r? vanish over r = a' ; but this implies that a', even when 
infinitely small compared to a, is still so great compared to mole- 
cular distances, that the action of molecules separated by a 
distance of order a' is inappreciable. There is thus no sudden 
discontinuity, as our solution seems to imply, but a gradual trans- 
ition as a increases from being a molecular distance to being a 
distance so great that the mathematical conditions for a free 
surface are satisfied. This transition stage is not within the 
compass of the present mathematical theory ; but this is hardly a 
matter of practical importance, for the mathematical conditions 
are probably satisfied in any existing hollow cylinder as exactly 
over its inner as over its outer surface. 

For brevity, a'ja = will be employed to denote the cylinder 
that is hollow in the mathematical sense, but in which a /a is 
extremely small. In the same way a /a = 1 will be employed to 
denote a cylinder whose wall thickness a — a is extremely small 
compared to a, though great compared to molecular distances. 

§ 6. Returning to our solution we see that £» vanishes when 
7) = 0, so that all the surface conditions are then exactly satisfied. 
The solution in this case is thus complete and applies to circular 
cylinders of all shapes, to thin disks as well as to very long 
cylinders. When r\ is small «» is very small compared to the 
greatest stress w, and even when rj is \ the greatest value of «* 
bears to the greatest value of JJ a ratio which is ^ in a solid 
cylinder and not more than J^ in a hollow cylinder. When, how- 
ever, 7) is | the greatest value of «£ in a solid cylinder is one half 
the greatest value of JJ, and so may be by no means a small 
stress. Now in the case of the thin disk the stresses which failed 
to vanish over the edges were always very small compared to the 
largest stresses. Thus, to all appearance, our solution for long 
cylinders is not quite so satisfactory as that for thin disks unless 
Poisson's ratio be small. 

§ 7. A striking difference between the effects of rotation on 
thin disks and on long cylinders in which rj is not zero, is that 
whereas in the former case originally plane sections perpendicular 
to the axis of rotation become paraboloidal, in the latter case 



1892.] Mr Chree, On Long Rotating Circular Cylinders. 287 

such sections remain plane. Unfortunately this absence of curva- 
ture could hardly be observed except at the ends, where our 
solution cannot strictly be applied. 

The shortening of the cylinder per unit of length is given by 

( - $l/l) = ( _ w /z) = co 2 p (a 2 + a' 2 ) V /2E (9), 

with a' 2 = for the solid cylinder. Though it vanishes with r\, 
this is in ordinary materials an important alteration. Its magni- 
tude in terms of co 2 pa 2 /E — a quantity depending on the density, 
Young's modulus and the velocity at the outer surface — is re- 
corded in the following table for the value "25 of tj, for various 
values of a'ja : — 

Table I. 

Shortening of cylinder per unit of length, <r\ — 25. 

a'ja= -2 -4 -6 '8 1 

( - 81/1) + (to 2 pa 2 /E) = -125 13 145 17 "205 -25 

The entry under a /a = applies also to the solid cylinder. 
Since the shortening varies directly as ij, its amount in terms of 
&pa 2 \E may be at once written down for any other value of rj. 
Numerical measures of ( — 81/1) in two typical cases will be found 
in Tables IX. and XI. 

§ 8. The other displacements of most interest are the altera- 
tions 8a and 8a' in the radii of the two cylindrical surfaces. The 
ratios of these alterations to the original lengths are given in 
terms of co 2 pa 2 /E by the formulae 

(8a/a) -j- (co 2 pa 2 /E) = I {1 - v + (3 + y) a'*/a?} (10), 

{8a' /a') - {a?pa 2 /E) = £ {3 + V + (1 - v ) a? /a*] (11). 

Thus the radii of both surfaces are always increased. Taking 
<a 2 pa 2 /E as constant, the following table shews how these altera- 
tions of the radii vary with rj and with a /a: — 

Table II. 

Value of (8a/a) - (co 2 pa 2 /E). 



7] a /a — 





•2 


•4 


•6 


•8 


10 





•25 


•28 


•37 


•52 


•73 


10 


25 


1875 


•22 


•3175 


•48 


•7075 


10 


5 


125 


16 


•265 


•44 


•685 


10 



288 Mr Ghree, On Long Rotating Circular Cylinders. [Feb. 8, 









Table I 


II. 












Value of {ha' fa') -. 


- (a>*pa*/E). 






7} a /a = 





•2 


•4 




•6 


•8 


10 





•75 


•76 


•79 




•84 


•91 


10 


■25 


•8125 


•82 


•8425 




•88 


•9325 


1-0 


'5 


•875 


•88 


•895 




•92 


•955 


1-0 



The numerical results in these two tables are exact*. Both ha fa 
and ha' ja' are linear in rj ; thus results for other values of w are 
easily and accurately supplied by interpolation. The results 
under a' fa = apply also to the solid cylinder in Table II., but 
not of course in Table III. The formulae (10) and (11) are 
identical with (45) and (46), Proceedings, I.e. p. 213, which give 
ha and ha' for a thin disk. Thus Tables II. and III. apply also to 
thin disks. 

The large and steady increase in the value of ha/a as a'/a 
increases is very conspicuous. It is also noteworthy that when 
a'/a has a given value, ha' /a increases but ha/a diminishes as 77 
increases. In fact by (10) and (11) 

ha/a -h ha' /a = (to a P a*/E) (1 + a' 2 /a 2 ), 

so that ha/a + ha' /a' is independent of 77. When a'/a approaches 1 
the influence of r\ on the magnitude of the alterations in the radii 
tends to disappear. An idea of the numerical magnitude of 
ha/a and ha' /a' will be most easily derived from the special cases 
treated in Tables IX. and XI. 

For the alteration ha — ha in the wall thickness we have 
the same formula as for a thin disk, viz. (47), p. 213, and this 
thickness is increased or diminished by rotation according as 

a'/a < or > (1 — Jn) + (1 + Jn) ; 

see Proceedings, I.e. Equation (48), p. 213, and subsequent remarks. 

Comparing Tables I., II. and III. it will be seen that ( — hl/l) 
is by no means negligible compared to ha/a and ha' fa unless 77 
be small. Thus as afl is small, the shortening of the cylinder 
should in general be more easily detected than the alterations 
in its radii. 

§ 9. Since ™ is zero the principal strains are everywhere 

dw du -, u 
-v— , -7- and - . 
dz dr r 

* This term as applied to this and following tables means that the numerical 
results are as exact as the formulae, and not merely the first figures of a decimal. 



1892.] Mr Chree, On Long Rotating Circular Cylinders. 289 

The longitudinal strain, -r- , is everywhere negative, i.e. a 

compression. It has the same constant value as wjz, and so 
is given in Table I. The transverse strain, u/r, is everywhere 
positive, i.e. an extension, and is never algebraically less than 

ft If 

the radial sti'ain -r- . Its greatest value, s. which is found at 

dr ° 

the axis of a solid cylinder or the inner surface of a hollow 
cylinder, is thus the greatest strain. For a hollow cylinder it 
is the quantity 8a' /a' of equation (11) and Table III. In a solid 
cylinder it is given by 

'-<**-3*&*j ^ 

whence answering to 77 = 0, 77 — "25 and 77 = 5 we obtain 

s + (co 2 pa 2 /E) = -37o, -2916 and '125 respectively. 

The radial strain is most conveniently dealt with by means 
of the formula 

8E(l-fj) du 

<o 2 p T dr JKX) { h 

where for a solid cylinder 

r~ z f (r) = a 2 (3 - 5 V ) - Sr (1 - 2 V ) (1 + 77) (14a), 

and for a hollow cylinder 

f(r) = - a 2 a" (1 + 77) (3 - 277) + (a 2 + a' 2 ) r 2 (3 - 577) 

- 3r 4 (1 - 277) (1 + 77) (146). 

For the sign of -=- we need only consider that of /(r). 

§ 10. In a solid cylinder f(r) is positive inside and negative 
outside the surface 

rW (3 -577) -{3 (1-277X1 + 77)} (15); 

but the radius of this surface exceeds a when 77 > 3. 

Thus in a solid cylinder when 77 > - 3 the radial strain is every- 
where an extension ; when, however, 77 < '3 there is a cylindrical 
surface, viz. (15), outside of which it is a compression. When 
77 = or *3 the radial strain vanishes over the surface of the 
cylinder, and elsewhere is an extension ; but for intermediate 
values of 77 the region wherein this strain is a compression has 
a small but finite thickness. For a given value of a this thick- 
ness has a maximum value of "03775a approximately when 77 = "2. 



290 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8, 

(mUj 

§ 11. The variations in the sign of -=- in a hollow cylinder 

may be most easily investigated by means of the equation 

f(r) = (16), 

where f(r) is given by (146), regard being had to the sign of 
the surface values of f(r), viz. : 

f(a') = - 2 v a* {(3 - V ) a 2 + (1 - S v ) a" 2 } (17), 

f(a) = - 2 v a 2 {(1 - 3t?) a 2 + (3 - v) «' 2 } (18). 

Let us denote by aj/a the least positive root, when real, of 

(x 2 + l) 2 (3 - 5 v f - \2x l (1 + v ) 2 (1 - 27?) (3 - 2 V ) = 0. . .(19), 

and by a t '/a the positive root of 

x 2 =(S v -l)f(S- v ) (20). 

Then a^ja is that value of a'ja for which f(r) has equal roots, 
and a t 'ja is that value for which f(a) — 0. When a'ja<a^ja 

then f(a) — and so -j- at the outer surface — is positive. It is 

obvious that (a')~ 2 /(a') — and so -j- at the inner surface — is 

negative for all permissible values of t) greater than 0. For the 
limiting case a'ja infinitely small, one root of (16) is of order a. 
This root is given, neglecting higher powers of a'ja, by 

r> 2 = (a> 2 ) (1 + V ) (3 - 2 v )/(3 - 5 V ) (21). 

Though for shortness we refer to this case as that where 
a'ja = 0, it is most convenient not to neglect the vanishingly 

small thickness i\ — a' within which -7- is negative, as we are 

thus enabled to include this case under the general classification. 
The phenomena may then be grouped under four classes, according 
to the value of 97, with transition cases. 

Class L, 77 = 0. 

Here (a')~ 2 f(a')=0=f(a), and f(r) is positive for all inter- 
mediate values of r. Thus for all values of a'ja, the radial strain 
vanishes over both surfaces of the cylinder, and elsewhere is an 
extension. 

Class II., < rj < -3. 

Here a[ja is imaginary. 

Sub-class (i), a'ja < a^'/a : 



1892.] Mr Ghree, On Long Rotating Circular Cylinders. 291 

f(r) vanishes over two distinct surfaces within the cylinder. 
Between these the radial strain is an extension, elsewhere it is 
a compression. 

Transition case, a'ja = a 2 '/a : 

The two surfaces over which f(r) vanishes coincide. At this 
surface the radial strain vanishes, elsewhere it is a compression. 

Sub-class (ii), a' /a > a 2 '/a : 

The radial strain is everywhere a compression. 

Transition case to Class III., rj = "3 : 

This follows the same laws as Class II., except that for a' /a = 
one of the two surfaces over which f(r) vanishes coincides with 
the outer surface of the cylinder, so that there is not, as in sub- 
class (i) above, a volume of finite thickness at the outer surface 
wherein the radial strain is a compression. 

Class III., - 3 < rj < (4 — j7)/3, i.e. -4514 approximately. 
Here a-[\a is real. 

Sub-class (i), a fa < a t '/a : 

f(r) vanishes over one surface within the cylinder, and the 
radial strain is a compression within, an extension outside of this 
surface. 

Transition case, a' /a = a-^ja : 

f(r) vanishes over a second surface, but this coincides with 
the outer surface of the cylinder. 

Sub-class (ii), a//a< a'/a< a 2 '/a : 

f(r) vanishes over two distinct surfaces within the cylinder. 
Between these the radial strain is an extension, elsewhere it 
is a compression. 

Transition case, a'ja = a 2 '/a : 

The two surfaces over which /'(>') vanishes coincide within 
the cylinder. At this surface the radial strain vanishes, elsewhere 
it is a compression. 

Sub-class (iii), a'ja > a 2 '/a : 
The radial strain is everywhere a compression. 
Transition case to Class IV. 
7] = (4 — v7)/3 ; a t '/a = a 2 '/a = '3728 approximately. 

This follows the same laws as Class III., except that Sub-class 
(ii) is not represented, and for a /a = '37 28 the radial strain 
vanishes over two surfaces both coinciding with the outer surface 
of the cylinder. 



292 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8, 

Class IV., (4 — n/7)/3 < v > 5. 

Sub-class (i), a /a < a^ja : 

f{r) vanishes over one surface within the cylinder, and the 
radial strain is a compression within, an extension outside of this 
surface. 

Transition case, a la = a-[\a : 

The radial strain vanishes over the outer surface of the 
cylinder and elsewhere is a compression. 

Sub-class (ii), a /a > a{\a : 

The radial strain is everywhere a compression. 

§ 12. As illustrating Class II. and the transition to Class III. 
we shall consider the values "25 and "3 of rj. The approximate 
values of a 2 '/a in these two cases are respectively '4276 and 
•3728*; i.e. the radial strain is everywhere a compression when, 
7) being '25, a'ja exceeds "4276, and when, rj being "3, a'ja exceeds 
"3728. When a'/a<a^/a the radial strain vanishes within the 
cylinder over two surfaces whose radii r t and r 2 are the positive 
roots of (16). The values of rja and rja for a series of values of 
a'ja are given in the following table to three places of decimals: — 

Table IV. 

Radii of surfaces over which -r- = ; rj = "25 and "3. 

a'/a= -1 -2 -3 -4 



134 -273 -423 616 

fa= -966 -962 -947 -916 '839 



7?=25 K/a= -966 ; 

. A irja = l'528a'/a '154 315 "5 

v a \rja = 1-0 -993 "970 '9165 



The radial strain is a compression at points whose axial 
distance lies between a' and r, or between r 2 and a, an extension 
where the axial distance lies between r x and r 2 . 

§ 13. As illustrating Class III. we shall consider the value *4 
of rj. For this we find approximately 

a; ja = -27735, aja = "3486. 

Thus when a /a is less than *27735 the radial strain vanishes 
over only one surface within the cylinder, being a compression 
within, an extension outside of this surface. When a'ja — "27735 
the radial strain vanishes over two surfaces, but one of these is 
the outer surface of the cylinder. W T hen a'/a lies between "27735 

* Equation (19) has the same roots when 77 = -3 as when t? = (4-\/7)/3. 



1892.] Mr Ghree, On Long Rotating Circular Cylinders. 293 

and "3486 the radial strain vanishes over two distinct surfaces, of 
radii r t and r 2 , within the cylinder, being an extension between 
these surfaces, and elsewhere a compression. When a/a= *3486 
these two surfaces coincide, so that the radial strain is nowhere 
an extension. Lastly, when a'/a is greater than *3486 the radial 
strain is everywhere a compression. The radii of the surface or 
surfaces where the radial strain vanishes are given approximately 
in the following table for the values 0, "1, "2 and '3 of a fa: — 

Table V. 

Radii of surfaces over which , = ; rj = 4. 

a'/a= -1 -2 -3 

rJa = V755a'/a 177 -364 -589 
rja = -975 

§ 14. As illustrating Class IV. we shall consider the limiting 
value o of i). For it we find approximately 

Oa'fa = -4472. 

Thus when a /a is Jess than "4472 the radial strain vanishes 
over one surface, r = i\, within the cylinder, being a compression 
within, an extension outside of this surface. When a'/a = "4472 
the radial strain vanishes over the outer surface of the cylinder 
and elsewhere is a compression. Lastly when a'/a exceeds *4472 
the radial strain is everywhere a compression. The approximate 
values of rja for the values 0, "1, "2, *3 and "4 of a /a are as 
follows : — 

Table VI. 

Radius of surface over which -=- = ; rj = "5. 

a'/a= -1 -2 -3 *4 

rja = 2-4i4i9a'/a "244 "480 '704 -910 

§ 15. The expressions for the stresses call for but little 
remark. In the solid cylinder »r vanishes over the cylindrical 
surface, and the same is true of h> when i] = '5. Elsewhere these 
stresses are always positive, i.e. tensions. The third principal 
stress, «», vanishes everywhere when rj = ; but for other values 
of r} it vanishes only over the cylindrical surface 

r = a/72, 

being a tension inside, a pressure outside of this surface. It is 
easily shown that rr is never algebraically greater than w> nor 
algebraically less than 7z. Thus at every point w> — 7z } = S, is a 



294 Mr Ghree, On Long Rotating Circular Cylinders. [Feb. 8, 

correct measure of the stress-difference. Its greatest value, 8, the 
maximum stress-difference, is found in the axis, being given by 

S = §a>>pa*{l-i V l(l- V )} (22). 

In the hollow cylinder it is obvious from (4) that rr vanishes 
over both cylindrical surfaces and elsewhere is a tension. Also w 
is always algebraically greater than 7? and so is necessarily a 
tension, though when r; = "5 it may be vanishingly small. The 
third principal stress 7z is never algebraically greater than m>. It 
is a tension inside a pressure outside of the surface 

r 2 = (a 2 + a' 2 )/2, 

where rr is a maximum. The stress- difference may be w — rr or 
H> — 7s according to the axial distance of the point considered, but 
the maximum stress-difference is always the value of h, - rr at the 
inner surface, and is given, since 7? is there zero, by 

B = $$r=a> = »> 2 {3 - 2 V + (1 - 2 V ) a"/a 2 } + 4 (1 - v ). . .(23). 

§ 16. In such a problem as the present the question perhaps 
of most practical importance is the determination of the greatest 
safe speed. Under ordinary conditions this is found according to 
the stress-difference theory by attributing to 8 a limit found 
experimentally ; on the greatest strain theory s is the quantity to 
which an experimental limit is assigned. I have elsewhere* 
discussed this question, pointing out that these theories at best 
can do no more than indicate the limiting stress or strain con- 
sistent with the stress-strain relations in the material remaining 
linear, i.e. obeying Hooke's law. There is, however, no mathe- 
matical objection to their application to determine a safe working 
limit, provided this is consistent with the linearity of the stress- 
strain relations. In the present case the question is complicated 
by the possibility of the motion becoming unstable. It is 
obvious, in fact, that in a long thin cylinder there is a danger 
that the axis under rotation may cease to be straight and may 
describe a spindle-shaped surface of revolution about the line 
joining its ends. This has been pointed out by Professor Green- 
hill -J", who has found formulae for the limiting speed, as depending 
on this kind of instability, in terms of the material and dimensions 
of the cylinder. 

I thus propose in the first place to give tables whence the 
limiting speeds allowed by the stress-difference and greatest 
strain theories may be obtained, illustrating their application to 
special cases ; secondly, to give data showing how it may be deter- 

* Phil. Mag. September, 1891, pp. 239—242. 

t Institution of Mechanical Engineers, Proceedings, 1883, pp. 182 — 209, with 
discussion, pp. 210 — 225. 



1892.] Mr Ghree, On Long Rotating Circular Cylinders. 295 

mined whether these or the speeds assigned by Professor Green- 
hill's theory are the lowest ; and thirdly, to direct attention to 
some details of practical interest. 

§ 17. All the methods of determining the limiting speed 
agree in the conclusion that in cylinders of the same material, in 
which Ija and a/a have given values, the limiting speed is 
reached when the velocity aa at the outer surface attains a 
certain value. Thus coa and not a> is the most convenient 
quantity to tabulate. 

On the stress-difference theory we assign to S in (22) and (23) 
a limiting value determined for each material by experiment. 
The following table is calculated from these formulae, the limiting 
velocity being termed co t a for the sake of reference. In using the 
table 8 has to be replaced by the experimental limit found for the 
material under consideration, and p by its density. 







Table 


VII. 












Value of a> x a -f- V S/p . 








Solid 
^ cylinder 


a /a = 


•2 


•4 


•6 


•8 


10 


1-633 


1155 


1-147 


1125 


1091 


1-048 


10 


•25 1-732 


1-095 


1-091 


1-078 


1-058 


1-031 


10 


•5 20 


10 


10 


10 


1-0 


10 


1-0 



When less than three decimal figures occur the result is exact. 

On the greatest strain theory the limiting speed is found from 
(12) and (11) — as s = ha'ja' in the hollow cylinder, — by assigning 
to s an experimental limit. The value of s like that of E or 
p varies of course from one material to another. Denoting by &> 2 a 
the limiting speed allowed by this theory, we obtain the following 
results : — 

Table VIII. 

Value of co 2 a -r jEsjp. 



Solid 
^ cylinder 


a fa = 


•2 


•4 


•6 


•8 


1-0 


1633 


1155 


1147 


1125 


1-091 


1-048 


10 


25 1-852 


1109 


1104 


1-089 


1-066 


1-036 


10 



•5 2-828 1-069 1066 T057 1-043 1-023 1-0 

All the results are only approximate except those in the last 
column. 



296 Mr Ghree, On Long Rotating Circular Cylinders. [Feb. 8, 

§ 18. In comparing the results of Tables VII. and VIII. it 
should be noticed that in a given material 8 = Es, if the two 
theories really apply to all forms of strain, because in a bar under 
simple longitudinal traction, s would be the longitudinal strain 
answering to a traction S. 

It thus appears that the two theories are in exact agreement 
when 7) = for all values of a'ja, and also when a' fa = 1 for all 
values of tj. Also while differing in details they both lead to the 
conclusion that as ij increases, other properties of the material 
being supposed unaltered, the safe speed rises in a solid cylinder 
but falls in a hollow cylinder for all values of a' fa; and further 
that in a hollow cylinder of given material as a' '/a increases, a 
remaining constant, there is a steady though not large fall in the 
safe speed. To this last law the stress- difference theory recognises 
an exception in the limiting case tj = "5, when the safe speed is 
according to it the same for all values of a' fa. 

The most striking result is unquestionably the great fall in the 
safe speed which according to both theories follows the removal of 
a core however thin it may be, consistent of course with the mathe- 
matical conditions for a free surface being satisfied. The magni- 
tude of this fall is the more conspicuous the larger n is. Even for 
7] = it amounts to over 29 per cent., and for t] = '5 it amounts on 
the greatest strain theory to over 62 per cent. For tj = "25, 
which is at least a fair approach to what is found in ordinary 
isotropic materials, the mean of the falls in the safe speeds pre- 
scribed by the two theories is approximately 38^- per cent. 

While the precise magnitude of the reduction in the safe speed 
due to the removal of a thin axial core may be questioned by 
those who regard with distrust existing theories of "rupture", the 
fact that there is a large reduction must I think be admitted by 
all who recognise the validity of the present solution, provided the 
safe speed is really regulated by the elastic state of the material. 
For our formulae show a large increase in the greatest values of 
every stress and strain to follow the removal of a thin core ; so the 
material can hardly fail to be brought considerably nearer to that 
critical condition where the stress-strain relations cease to be 
appreciably linear, whatever the precise elastic quantity may be 
on which that condition depends. 

§ 19. To illustrate the use of the previous tables, and to give 
an idea of the range of the numerical magnitudes of the several 
quantities tabulated, I shall now consider some special cases. 

The first case, to which Table IX. refers, is designed to show 
the magnitudes of the principal displacements and greatest strains 
when the maximum stress-difference is of given magnitude. The 
value of 77 in the material is taken to be "25. Since (— 81/1), 8a fa 



1892.] Mr Ghree, On Long Rotating Circular Cylinders. 297 

and s, when ij is given, vary simply as S/E, it is most convenient 
to attach a value to this ratio and not to 8 itself. This has the 
further advantage that S/E is a purely numerical quantity, inde- 
pendent of the system of units adopted. It is taken in Table IX. 
to be '001. This value is selected principally owing to the facility 
with which it lends itself to the deduction from the table of 
numerical values in other special cases. It is not intended to 
imply that this is the true ratio of the greatest allowable stress- 
difference to Young's modulus in any actual material. 

Table IX. 
S/E = -001 ; v = -25. 

n S v h A a'/a = 2 -4 "6 '8 10 

Cylinder ' 

(-81/1) x 10 s = S7 5 150 155 169 190 -218 25 

O/a)xl0 3 = -562 -225 -262 -369 '537 -753 1-0 

§xl0 3 = -875 -975 -976 -980 -985 '992 1-0 

The results, with one or two exceptions, are only approximate. 
In the hollow cylinder s = Sa'/a', so the table gives also the increase 
in the radius of the inner surface. 

§ 20. The velocity ma, as may be seen by reference to equa- 
tions (22) and (23) or to Table VII., varies as Js/p. So in calcu- 
lating it the absolute values of S and p, or rather their ratio, and 
not the value of S/E, is wanted. 

Since British engineers seem accustomed to measure velocity 
in feet per second and stress in tons weight per sq. inch, these 
units have been adopted in Table X. Two special cases are there 
dealt with, in both of which rj is taken as "25. The first case, in 
which the velocity is styled w^a, answers to 8 = 12 tons wt. per 
sq. inch, p — 7*5 times the density of water. This selection is 
made so as to fit in with the case treated in Table IX. For if, as 
there, we suppose S/E—'OOl, then E = 12 x 10 3 tons wt. per sq. 
inch, i.e. approximately 18'90 x 10 8 grammes wt. per sq. cm. Now 
this value of E and a specific gravity of 7 "5 may fairly be taken as 
representing steel or wrought iron, though rather low values for 
good material. The second case in Table X., where the velocity is 
styled ro/a, answers to 8 = 1 ton wt. per sq. inch, p = the density 
of water, or more generally to 8 — n tons wt. per sq. inch, 
p = n times the density of water. This selection is made with a 
view to facility of application to other special cases. The results 
are all approximate. 



298 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8, 

Table X. 
Velocity in feet per second; 77 = 25. 



Solid , 
Cylinder 


ja = 


•2 


•4 


■6 


•8 


10 


tojtt = 893 
a,/a = 706 


565 

447 


563 
445 


556 
440 


546 
431 


53£ 
421 


516 

408 



§ 21. The next special case, that treated in Table XL, sup- 
poses the greatest strain s=*001, while 77 = "25 as before. These 
are the only data on which (— 81; I) and 8a 'a depend ; these quanti- 
ties, when s is given, being independent of E or p. 



Table XI. 

s=-001; 7] =-25. 

n S ° h j ala = -2 -4 6 -8 TO 

Cylinder 

(-SZ/Z)xl0 3 = -429 154 159 -172 -193 -220 -25 

(Ba/a) x 10 3 = -643 -231 -268 -377 "545 -759 1-0 

The results with the exception of those in the last column are 
only approximate. 

In calculating S we require in addition to s the value of E, but 
not that of p. In Table XII. the value assigned to E is 20 x 10 8 
grammes wt. per sq. cm., which is a fair average for good wrought- 
iron or steel. For practical convenience the maximum stress- 
difference is given in tons wt. per sq. inch, taking 703083 grammes 
per sq. cm. as equal to 1 lb. per sq. inch. The results are all 
approximate. 

Table XII. 
s = '001 ; E — 20 x 10 8 grammes wt. per sq. cm. ; tj = "25. 

n S r h t a? 1(1 = '2 -4 '6 "8 TO 

Cylinder 

S, in tons J = 14 . 51 13-025 13-01 1296 12-89 12-80 12-70 
per sq. inch] 

In calculating the velocity we require in addition to the values 
of s and E the value of p. In Table XIII. it is takeD to be 7*5 
times the density of water, while s and E have the same values as 
in Table XII. The velocity is termed com for distinction ; it is 
measured in feet per second. The results are all approximate. 



1892] Mr Chree, On Long Rotating Circular Cylinders. 299 

Table XIII. 
s = '001; £" = 20 x 10 s ; p = 7'5: v = 25. 

n S , oll f a'/a = Q 2 -4 -6 S 10 

Cylinder ' 



531 



pe^^condV 983 589 586 578 566 550 

In comparing the oo. 2 a of Table XIII. with the com of Table X. 
it must be remembered that the values assigned to E in the two 
cases are different. The proper basis of comparison may at once 
be arrived at from the fact that the stress-difference and greatest 
strain theories are in exact agreement in the limiting case 
a/a = 1. 

§ 22. The results in the last three tables may easily be 
applied to other special cases when 77 = - 25. Since ( — 81/1) and 
Sa/a vary simply as s, if any value other than - 001 be assigned to 
s we have only to alter the numbers in Table XI. in the same pro- 
portion. We may also make use of the facts that $ varie3 
directly as Es, and com varies as J Es/p, to adapt Tables XII. and 
XIII. to other cases. 

For instance, let us take flint-glass, assuming for it 7) — '2o and 
allowing to s the value "0008. Let us take E = 6 x 10 8 grammes 
wt. per sq. cm., and p = 294 times the density of water, values 
which are approximately the mean of those given by Professor 
Everett*. Then to obtain numerical results for this case we have 
only to multiply the values of (— 81/1) and 8a/a in Table XI. by - 8, 
the values of S in Table XII. by (6/20) x '8 or "24, and the values 
of com in Table XIII. by J(6/20) x "8 x (7\5/2-94) or '7825 
approximately. 

§ 23. We have next to consider the limits which Professor 
Greenhill has found for the safe speed when none but "centrifugal" 
forces are supposed to act. He imagines the cylinder bent under 
rotation so as to be in equilibrium under the "centrifugal" forces, 
which tend to increase its bending, and the elastic forces whose 
tendency is to keep it straight. He thence . arrives at a relation, 
varying with the terminal conditions, between the velocity of 
rotation and what is practically the ratio of the length to the 
diameter of the cylinder. Taking the velocity and cross section as 
given, Professor Greenhill regards this as fixing the greatest length 
of cylinder in which rotation is stable. 

* Units and Physical Constants, 2nd Edition, Art. 64. 
VOL. VII. PT. VI. 24 



300 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8, 

The method is analogous to that whereby we obtain Euler's 
formula for the greatest length permissible in a strut of given 
section subjected to a given load. If instead of fixing the velocity 
of the cylinder we suppose the ratio of length to diameter given 
us along with the cross section, Professor Greenhill's relation 
assigns a limiting speed which cannot be exceeded without ex- 
posing the cylinder to buckling. 

The physical problem treated by Professor Greenhill is one of 
great difficulty, and doubts may be entertained as to how closely 
it is reproduced in the mathematical problem which he has solved. 
This question would however lead us too far afield, and we shall 
here consider merely the conclusions to which the instability 
formulae lead, throwing all responsibility for the exactness of the 
instability theory upon its author. 

§ 24. For the case of simple rotation Professor Greenhill 
gives two formulae*. In deducing the first he takes 

at the ends of the cylinder, where y denotes the perpendicular 
from a point on the axis, supposed bent under rotation, on the 
straight line coinciding with its undisturbed position. This sup- 
poses the axis constrained to retain its original direction at the 
ends. The terminal conditions assumed in the second formula 
are 



y 



" TO* 



d%*' 



This answers to zero curvature of the axis at the ends, or on 
Professor Greenhill's interpretation of the problem to the vanishing 
of the elastic bending couple. 

The conclusions to which the formulae lead appear at first 
sight widely different, the length allowed by the first set of 
terminal conditions in a cylinder of given section rotating with a 
given speed being according to Professor Greenhill 9"46/7r times — 
or approximately thrice — that allowed by the second. The differ- 
ence is however in considerable part due to a slight slip made by 
Professor Greenhill in attaching a numerical value to a quantity 
he terms \id. For this he quotes (I.e. p. 199) the value 4"73, 
whereas in reality 

\ld = 2-36502, 

or only half this value. 

* Those numbered (19) and (20) in Professor Greenhill's paper, pp. 199 and 200. 



1892.] Mr Chree, On Long Rotating Circular Cylinders. 301 

Amending this numerical coefficient, and employing as pre- 
viously 21 for the length of the cylinder, in place of Professor 
Greenhill's I, we may represent his two results under the forms 

co'p/Ek' 2 = (2-36502/0 4 (24), 

co'p/EK 2 = (7r/2iy (25), 

where k denotes the radius of gyration of the cross section about 
a perpendicular through its centre to the plane of bending. The 
rest of the notation has its previous signification. Professor 
Greenhill apparently introduces no restriction as to the nature of 
the cross section ; thus the formulae apply presumably to hollow 
as well as solid cylinders. 

The terminal conditions assumed in (25) are based on the 
elastic theory adopted by Professor Greenhill; while (21) merely 
assumes the direction of the axis fixed at the ends, and so seems 
exposed to fewer uncertainties. I shall thus direct my attention 
to the first of the two formulae ; but results answering to the 
second can be easily derived from those answering to the first by 
replacing the factor (118251) 4 in equations (26) to (29) below by 
(tt/4) 4 . 

§ 25. Let us then suppose that in a certain isotropic solid 
circular cylinder the limiting velocity as prescribed by (24) is such 
as to cause in the material a maximum stress-difference S. Then 
remembering that a 2 /4 is the value of k 2 in a circle, and that the 
angular velocity which appears in (24) is the same as appears in 
(22), we immediately deduce 

8/E = (ri8251) 4 (ft/Z) 4 (3 - 4?) -5- 2 (1 - v ) (26). 

Similarly if 5 be the greatest strain answering to the velocity 
prescribed by (24) we find by means of (12) 

s = (l-18251) 4 (ci/iy(Z-5 v ) + 2(l- v ) (27). 

In a circular annulus k 2 = (a 2 + a' 2 )/4. Thus, remembering 
that s = ha'ja', we find from (23) and (11) for the maximum stress- 
difference and greatest strain in a hollow cylinder, answering to 
the limiting velocity prescribed by (24), the respective values : 

SfJS = (1-18251) 4 (a/iy (1 + a' 2 /a 2 ) x 

(3_2 7? + (l-27 7 ) a > 2 }-(l-7 ? ) ....(28), 

5 = (118251) 4 (a/iy (1 + a' 2 /a 2 ) [3 + V + (1 - v ) a 2 /a 2 } . . .(29). 

§ 26. It is obvious that the maximum stress-difference and 
greatest strain answering to the limiting velocity prescribed by 
the instability formula always diminish very rapidly as the ratio of 
the length of the cylinder to its diameter increases. Also when 

24 2 



302 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8, 

the magnitude of S/E or s is known for any one value of a/l, it is 
a simple matter to obtain its magnitude for any other value of ajl 
supposing 77 and a'/a unaltered. The following table gives the 
approximate numerical magnitudes of S/E and 5 when I = 10a, 
the value of 77 in the material being "25. 



Table XIV. 



Solid 



a'ja 



cylinder '2 -4 '6 -8 10 

(£/#) x 10 3 = -261 -652 -683 780 -950 1-206 1-564 
8 x 10 s = -228 -635 -667 -764 -936 1-196 1*564 

§ 27. By assigning a given value to S/E or s in equations 
(26) to (29) we obtain a value of l/a which gives the shortest 
cylinder in which so large a value of S/E or s would be reached 
when the velocity was kept within the range prescribed by the 
Greenhill formula (24). If this value of S/E or s be the extreme 
limit of safe working in the material considered, the corresponding 
value of l/a determines the shortest cylinder in which the safe 
speed can properly be assigned by (24). Suppose for instance 
t] = 0, the case in which the stress-difference and greatest strain 
theories agree, and let S/E = s = -001 denote the safe working 
limit, then the values of l/a for the shortest cylinders in which it 
is legitimate to determine the safe speed by means of (24) are 
those given in the following table : — 



Solid 



Table XV. 

a'/a 



cylinder -2 '4 *6 -8 TO 

l/a = 7-36 8-75 8-87 920 9 72 1039 1118 

For the limiting case a'/a = 1 the result is independent of 77, 
and in fact the value of 77 is of comparatively little importance 
except in the case of the solid cylinder or when a'/a is small. 
The difference between the limiting values of l/a allowed to the 
application of (24) in a solid cylinder and in a hollow cylinder in 
which a'/a is vanishingly small is least when 77 = 0. It is greatest 
when 77 = "5, in which case we find for l/a in a solid cylinder 
answering to S/E ='001, and to s=*001 the respective values 
6'65 and 5-59; while for the hollow cylinder a'/a = the cor- 
responding values are no less than 940 and 910. 



1892.] Mr Chree, On Long Rotating Circular Cylinders. 303 

Since l/a according to the formulae (26) to (29) varies as 
(E/S)* or (1/s)* it is easy to adapt results such as those of 
Table XV. to cases when values other than - 001 are allowed 
to S/E or s. A caution may however be not unnecessary as 
to the use of such results, viz. that unless r\ vanish or be very 
small the solution obtained in the present paper is not altogether 
trustworthy when l/a is markedly less than 10, and that the 
Greenhill theory is probably in such a case still more untrust- 
worthy. 

§ 28. Tables XIV. and XV. show that Professor Greenhill's 
formula (24), assuming his theory trustworthy in itself, ought 
not to be applied in determining the limiting speeds in hollow 
cylinders whose length is less than 12 or 13 times their diameter, 
without a check being applied by reference to the results of 
the present paper. The thinner the walls of the cylinder the 
more necessary is the check. In solid cylinders, a check of this 
sort is less necessary, but it must be remembered that when 
l/a is less than 10 the hypotheses on which (24) is based can 
hardly be considered satisfactory. 

§ 29. While the exact law laid down by the Greenhill for- 
mula, viz. that the limiting speed a>a as far as instability is con- 
cerned varies as (a/If, cannot be relied on in short cylinders, 
there can be little doubt that there is a continuous and rapid 
diminution in the tendency to instability as l/a is reduced 
below 10. On the other hand, a large increase in the limiting 
speed allowed by the stress-difference and greatest strain theories 
can hardly accompany this reduction of l/a. 

For while in short cylinders the strains and stresses will doubt- 
less for ordinary values of tj vary appreciably with z, the mean 
value of a certain strain or .stress-difference for a given value 
of r when taken between + I seems hardly likely to vary much 
with l/a. 

Thus it is a priori improbable that the greatest values 
reached by these quantities in a short cylinder can be much 
less than the values attained in a long cylinder, the values of 
aa and a/a being the same, though conceivably in some cases 
they may be appreciably greater (cf. § 31). 

The following considerations afford strong support to this 
view : The expressions we have found for the maximum stress- 
difference and greatest strain both in thin disks and long cylinders, 
when coa, a /a, p and E are treated as constants, vary with the 
value of 7j only within comparatively narrow limits (see § 17). 
Unless then the influence of rj on the magnitude of these 
quantities be very much greater in cylinders of intermediate 



304 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8, 

length, a fair idea of this magnitude in any material may be 
derived by supposing t] to vanish. But our solution is exact 
when 77 = 0, and the maximum stress-difference and greatest 
strain are then wholly independent of Ija. 

There is thus, on various grounds, a strong presumption that 
in any isotropic material the limiting safe speeds allowed by a 
complete elastic theory in short cylinders would not differ very 
much from those which the elastic theory of the present paper 
allows in long cylinders. Thus, even if Professor Greenhill's 
method could be trusted when Ija ceases to be large, recourse 
would require to be had to the strict elastic theory and not to 
his formulae to fix a limiting speed. 

§ 30. In Professor Greenhill's paper, and in the discussion 
which follows it, reference is made to propeller shafts in steamers. 
These shafts, at least in large steamers, are of a length very great 
compared to their diameter. The "Servia" for instance is said 
to have a solid shaft 164 feet long and only 22^ inches in diameter. 
It seems however the practice to support the shaft at intervals ; 
in the "Servia" for example there are bearings making of the 
shaft 8 lengths. In such a case the distance between the 
bearings ought presumably to be regarded as the length 21 in 
applying the instability formula. The value of Ija may perhaps 
still be great enough for a legitimate application of the Greenhill 
theory, but it is by no means so great that a consideration of 
the magnitude of the elastic strains and stresses can be safely 
dispensed with, at least in wrought-iron. When a shaft of this 
kind is hollow, as is sometimes the case, the magnitudes of the 
elastic strains and stresses given by our formulae should be looked 
to, even if the material be the best steel. These remarks assume 
of course that none but " centrifugal " forces act. In propeller 
shafts under normal conditions this is far from true, there being 
torsional and longitudinal forces as well, whose effects may be 
large compared to those of the " centrifugal " forces. When 
such a compound system of forces acts, the limiting speed ac- 
cording to the elastic theory is to be got by superposing the 
displacements arising from the several sources, and then ascribing 
limiting values to the maximum stress- difference or greatest 
strain given by the complete solution. It would however be 
straying too far from our main subject to discuss this matter 
further. 

§ 31. Reference has already been made to the comparatively 
small difference between the limiting speeds allowed by the 
elastic theories of " rupture " in long cylinders and in thin disks. 
The exact relations between these speeds arc as follows: 



1892.] Mr Chree, On Long Rotating Circular Cylinders. 305 

Let the limiting speeds in a long solid cylinder and in a 
complete disk, both of radius a and of the same isotropic material, 
be denoted by w x and 12, respectively on the stress-difference 
theory, by &> 2 and fl 2 respectively on the greatest strain theory. 
Then by reference to my previous paper, we easily find 

ov/H^l + 77 (2-^/(3-4^) (30), 

<yn 2 2 = I+17 2 (1 + ^/(3 -5*7) (31). 

Hence eoj/fli and o)Jfl 2 both vary from 1 when w = to J 7/4, 
or 13229 approximately, when r/ = '5. For w = '25 we have 
approximately 

6)^ = 1-104, 6) 2 /n 2 = 1-022. 

For a hollow cylinder and an annular disk of the same 
isotropic material, with the same values of a and a, we find, 
distinguishing the results from those given above by dashed 
letters : 

(a^'/n/) 2 = 1 - V 2 (1 - a' 2 /a 2 ) -=- {3 - 2 V + (1 - 2?) a' 2 /a 2 } . . .(32), 
(<y 2 '/n 2 ') 2 = 1, for all values of 77 and of a /a (33). 

Thus the limiting speeds on the greatest strain theory are 
here identical, and on the stress-difference theory the difference 
between the speeds is extremely small for ordinary materials, 
especially when 1 — a'ja is small. It is worth noticing that 
according to the stress-difference theory the limiting speed is 
less in the long hollow cylinder than in the corresponding thin 
disk, whereas it is greater according to both theories in the long 
solid cylinder than in the complete disk of the same radius. 



Monday, Feb. 22, 1892. 
Professor G. H. Darwin, President, in the Chair. 

The following communications were made to the Society : 

(1) Some jyreliminary notes on the anatomy and habits of 
Alcyonium digitatum. By Sydney J. Hickson, M.A., Fellow 
of Downing College, Cambridge. 

Alcyonium digitatum is one of the most difficult Coelenterates 
to kill in a fully expanded condition. In the first place it is only 



306 Mr Hickson, Some preliminary notes on the [Feb. 22, 

extremely rarely that the large proportion of the polypes of a 
specimen in an aquarium fully expand themselves, and when they 
are in that condition the slightest touch or irritation of any part 
of the colony causes an immediate contraction of the tentacles. 
Again when a favourable opportunity arises it is found that all the 
neutral killing reagents such as corrosive sublimate, etc. fail to kill 
the polypes before they have time to partially retract. The only 
method that gives tolerably satisfactory results is Lo. Biancho's 
No. II. Chromo-acetic acid method, and this of course partially 
dissolves the calcareous spicules. 

When a living specimen of an Alcyonium digitatum is examined 
in an aquarium the polypes may frequently be observed in various 
stages of expansion and retraction. Sometimes all the polypes 
are completely retracted, but I have never yet observed in any 
specimen all the polypes fully expanded at the same time. By far 
the most frequent condition of the Alcyonium is one in which 
a few polypes here and there are fully expanded, others expanded 
but with their tentacles contracted, and others only just protruding 
from the surface of the colony. 

These two stages are the normal ones that each polype passes 
through in reaching complete retraction from complete expansion. 
When the polype is completely expanded both the body-wall and 
the tentacles are delicate and transparent. 

The first stage in the retraction is the contraction of the 
tentacles. The crown of the polype becomes roughly octagonal 
in shape with an obtuse solid knob — the contracted tentacle at 
each angle, 

In the next stage the contracted tentacles bend over towards 
the mouth and concurrently with the retraction of the body 
of the polype they sink into a circular fold of the body- 
wall. 

The invagination of the polype then proceeds at the base until 
the crown covered by the fold of body-wall sinks below the surface 
of the colony. 

When the crown has sunk below the surface of the colony the 
aperture is closed by the folding of the delicate body-wall of the 
base of the polype over the crown, but when the colony enters into 
a state of complete contraction, as it does for example when it is 
taken out of the water for a few minutes, the tough obtuse surface 
of the colony contracts over this delicate base leaving only a star- 
like slit to mark the position of the retracted polype. 

The ectoderm is in the uninjured specimens composed of 
several layers of cells. Under unhealthy conditions the superficial 
layers are apt to slough. 

The stomodaeum of each fully-developed polype opens into 
a long coelenteron that passes down to the base of the colony. 
In a longitudinal section through a branch of the colony the 



1892.] anatomy and habits of Alcyonium digitatum. 307 

coelentera spread from the base in a fan-like manner towards the 
periphery. 

Between the coelentera there is a dense clear mesogloea 
containing spicules, endodermic cell cords, a very delicate network 
of nerve (?) fibrils and cells. 

The young polypes originate in these endodermic cords, the 
coelentera being developed later, and do not communicate directly 
with the coelentera of the older polypes until they are nearly full 
grown. 

The endodermic cords are usually described as canals but there 
is no evidence of the presence of a lumen. Numerous injection 
experiments failed to prove the existence of any cavity in these 
structures. 

The nerve (?) netwoik can only be made out in fresh osmic 
acid preparations. It consists of a fine network of delicate fibrils 
connecting isolated mono- bi- and tripolar cells. It is difficult to 
trace in the peripheral parts of the colony, since the presence of 
a very large number of spicules makes it a matter of some diffi- 
culty to cut very thin sections of the fresh tissues. 

When the tide is low in the tropics some forms of Zoophytes, 
such as Tubipora, Clavularia, Sarcophytum, the Astraeidse and 
a few others, remain expanded until the water actually leaves the 
edge of the reef. Others on the other hand, such as Heliopora, 
Millepcra and most of the Madrepores, completely retract while 
there is still a foot or more of water covering them. Some 
Zoophytes in fact appear to anticipate low tide before others, and 
it occurred to me that this might be to a certain extent due to 
the development of a rhythm similar to the rhythmic movements 
of certain plants. 

In order to determine if possible the truth or falsity of this 
supposition I made last autumn a series of experiments upon 
Alcyonium digitatum in the tanks at the Plymouth laboratory. 

I placed a number of specimens of Alcyonium digitatum, 
collected partly in the shallow water of the Catwater and partly 
in deeper water off the Eddystone lighthouse, in one of the tanks, 
and I noticed that nearly all of them contracted completely twice 
in every twenty-four hours for the first three or four days. 

Those that did not contract in this manner soon showed 
bubbles of gas in their tissues and commenced to putrify. 

I also placed a number of specimens in another tank in which 
I arranged an artificial tide by means of siphons. It was so 
arranged that the water should run off — but not so completely 
as to leave the Alcyoniums uncovered — in twelve hours and fill 
up again in twelve hours. The Alcyoniums contracted with toler- 
able regularity twice in twenty-four hours for the first two days 
and then contracted quite irregularly, some only once, some twice. 
At the end of a fortnight two out of the three that remained in 



308 Mr Hickson, Some notes on Alcyonium digitatura. [Feb. 22, 

a healthy condition contracted regularly only once in twenty-four 
hours. 

These experiments appear to indicate 1st that there is a 
rhythmic contraction of the polypes of Alcyonium digitatum in 
the normal conditions twice in every twenty-four hours, 2nd that 
a new rhythm may be induced by an artificial tide of different 
duration to the natural one. 

On the other hand the results of the investigation are not 
altogether satisfactory on account of 1st the large number of 
specimens that became unhealthy during the progress of the 
experiment, 2nd the fact that the only three specimens that 
survived in the tank with the artificial tide were taken from the 
deep water off the Eddystone. 

Far more satisfactory results would probably be obtained if 
experiments were tried upon a number of Alcyoniums taken from 
shallow water direct and placed in the tanks. Many of the 
specimens collected from the Catwater were probably taken by 
the trawlers from deep water and thrown overboard as they entered 
the harbour and were in consequence in an unhealthy condition 
Avhen collected. 

The subject however seems to me to be worthy of further 
investigation. 



(2) On the action of Lymph in producing Intravascular 
Clotting. By Lewis Shore, M.D., Fellow of St John's College. 

From 1883 to 1889 numerous papers 1 were published by 
Wooldridge on intravascular clotting. He showed that when a 
watery extract of lymphatic glands, of testis, of thymus and of 
other tissues was injected into the blood, death by more or less 
extensive intravascular clotting was produced. He further showed 
that this property depended on the presence in the extracts of 
substances, proteid in nature, named by him tissue-fibrinogens. 
He could also obtain these bodies from chyle 2 , but he made no 
definite statement that ordinary lymph could produce intra- 
vascular clotting ; but it is clear from his paper on Auto-infection 
in Cardiac disease 3 that he attributed to it such power, although 
he had no definite experimental proof to bring forward. When 
we consider that he proved the power to reside in the juice 
expressed from lymphatic glands it seems very remarkable that he 
did not try lymph drawn from the thoracic duct. The various 

1 Chiefly in Proceedings of Royal Soc, and Journal of Physiology. 

- Ludwig's Festschrift, 1887. 221. 

3 Proceedings of Royal Society, xlv. 309. 



1892.] Mr Shore, On Intravascular Clotting. 3U9 

extracts and fluids he injected are all artificial, and none, not even 
the expressed juice from the lymphatic glands, can be considered 
as normally present in the body in exactly the form in which he 
injected them. It was therefore of the greatest importance from 
his point of view to show that the lymph as it is normally passing 
to the blood carries with it the very substances which the arti- 
ficial extracts contain, viz. tissue fibrinogens, and that the lymph 
itself when injected in the same way as the extracts, does by 
virtue of these bodies produce intravascular clotting. The present 
paper is a preliminary account of a few experiments I have so far 
made which supplement and continue Wooldridge's work on this 
point. The observations were made during the continuation of 
my work on the action of peptone on the clotting of Lymph and 
Blood 1 . It was shown by Heidenhain' 2 that when peptone is 
injected into the blood, the flow of lymph is much increased and 
the percentage of proteids in it increased. It was possible then, 
that the blood had lost to the lymph certain proteids necessary to 
clotting, and that the cause of the loss of clotting power was to be 
sought by studying the relations of the lymph to the blood. The 
addition of peptone lymph to peptone blood out of the body did 
not lead to clotting. I then turned to the intravascular injection 
of lymph, and first of all to normal lymph. 

The lymph was collected from the thoracic duct of a dog in 
the ordinary way. In the first experiment 12 c.c. of lymph could 
be collected before it clotted. This was quickly injected into the 
jugular vein of a rabbit, there was some rapid respiration, and in 
2 — 3 minutes the animal was dead. The heart, and all the 
arteries and veins, even their small branches, as far as traced, con- 
tained only clotted blood. The clotting was complete so that long 
" blood casts " could be drawn from the vessels. A smaller quan- 
tity of the same lymph, 5 c.c, was then collected and injected 
into another rabbit. In about one minute the animal was dead, 
and general intravascular clotting was found as before. It was at 
once considered possible that the difference in the species of the 
two animals might be of great importance. Richet and Hericourt 3 
found that 12 c.c. of defibrinated dog's blood was sufficient to kill 
a rabbit, but the condition found after death was not one of 
general intravascular clotting, as only a few small clots were 
formed, although there was a profound influence on the red 
corpuscles. I injected 15 c.c. of freshly drawn arterial blood from 
the same dog without producing any effect on the clotting power 
of the rabbit's blood, neither hastening or retarding clotting. 
Lymph from the same dog was allowed to clot, and 15 c.c. of the 
lymph serum, injected into a rabbit, also produced no effect. 

1 Journal of Physiology, xi. 561. " Pfliiger's Archiv, xlix. 209. 

3 Comptes rendus de la Soc. de Biol, cvn, 718 and ex. 1282. 



310 Mr Shore, On the action of Lymph [Feb. 22, 

Subsequent experiments showed that the intravascular clotting 
produced in the rabbit by the injection of the lymph of the dog is 
not always so complete as in the experiments just recorded. In 
many cases the animal died very soon after the injection was 
completed, but on examination clots were found only in the heart. 
In some cases the clots were restricted to the right side of the 
heart. Again in some cases the animals died within two or three 
minutes after the injection, and generally after some disturbed 
respiration, but the blood even in the heart was found to be 
fluid. Sometimes however in these cases a careful examination 
has revealed minute clots in the right side of the heart and in 
the pulmonary vessels. The blood in these cases usually clotted 
almost instantly it was shed. Several cases have occurred in 
which the injection of lymph produced no obvious effect on the 
rabbit. 

I am not able at present to fully explain this considerable 
variability in action. The explanation is to be sought rather in 
the condition of the lymph injected than in variations in the blood 
of the rabbit. It would be naturally expected that the variation, 
in the density, the amount of proteids, and the number of 
leucocytes, &c. which occur in the lymph of the dog, would 
influence the amount necessary to be injected to produce intra- 
vascular clotting. The rate of clotting of the lymph limits the 
amount which can be used, and it is rare that more than 10 c.c. of 
normal lymph can be collected before clotting has set in. This is 
a serious difficulty in the work, and in the negative cases observed 
a rapid clotting lymph was generally recorded. The most active 
lymph in producing intravascular clotting is that obtained from a 
dog in active digestion. Wooldridge also observed that the intra- 
vascular clotting produced by the injection of tissue-fibrinogen is 
more readily produced and is more extensive when the animal 
experimented with is in full digestion. I have however observed 
that the opaque white lymph in these cases is sometimes inert. 

Large quantities of lymph, which does not clot, may be 
obtained by the injection of peptone into the vascular system. 
Such lymph is however unable to cause intravascular clotting 
when injected into a rabbit. So also is "salted lymph," that is 
lymph received into solutions of neutral salts. That this can be 
explained by the presence in the lymph in these cases of sub- 
stances opposing the action of the clotting-exciting substance, is 
probably shown by the fact that tissue-fibrinogen can be separated 
from the plasma of such lymph, and that its injection leads to 
intravascular clotting. 

Tissue-fibrinogen was obtained from lymph in the following 
way. The lymph was allowed to drop directly from the thoracic 
duct into a very small quantity of very dilute acetic acid, a few 
more drops of the diluted acid being added from time to time as 



1892.] in producing Intravascular Clotting. 311 

the amount collected increased. A precipitate was thus at once 
formed, this was separated by centrifngalisation, and then twice 
washed with water as Wooldridge directed. On the addition of a 
small quantity of alkali, very dilute sodic hydrate, the precipitate 
almost completely dissolves. (It consists partly of leucocytes 
separated by the centrifuge.) 

The tissue-fibrinogen solution itself clotted in 2 — 5 minutes to 
a firm jelly. If injected at once into a rabbit, the animal dies 
almost immediately, and clots are found in the heart and in some 
cases the portal system. This result was however not always 
produced, sometimes the animal remained alive, but its blood was 
found to have lost its clotting power. This is in full accord with 
the observations of Wooldridge, with tissue-fibrin ogen obtained 
from glands and testis. 

A substance, which produces intravascular clotting, may be 
obtained from lymph by a method indicated by Alexander 
Schmidt 1 . He found that an alcoholic extract of lymphatic 
glands, of liver, and other tissues contained a substance which 
excited clotting in horse's plasma. I allowed lymph to drop from 
the thoracic duct directly into alcohol, the precipitate was then 
shaken with the alcohol for several hours, and then the alcohol 
was poured off and evaporated to dryness at 37°, and the residue 
shaken up with a little water. A considerable portion dissolved, 
and 4 c.c. of this, which formed a clear, slightly pinkish coloured 
solution, was injected into the jugular vein of a rabbit. The 
animal died at once, and the heart and large veins were full of 
clotted blood. The alcoholic extract of blood treated in the same 
way does not produce intravascular clotting. 



(3) On the fever produced by the Injection of sterilized Vibrio 
Metschnikovi cultures into rabbits. By E. H. Hankin, B.A., 
Fellow of St John's College, and A. A.' Kanthack (M.B. Lond.), 
St John's College. 

It is well known that a fever can be produced in rabbits by 
intravenous injection of the products of the growth of Vibrio 
Metschnikovi. a microbe closely allied to the Cholera-germ. The 
serum of rabbits possesses a certain power of killing the Vibrio 
Metschnikovi, and we have attempted to find out whether this 
' bactericidal power' is increased during or after the fever produced 
by the method above mentioned. We have also noted the numbers 
of leucocytes present in the rabbit's blood before and after the 
fever, in the hope of being able to establish some relation between 

1 Centralblatt fiir Physiologic, iv. 257. 



312 Messrs Hanlcin and Kanthack, On the fever produced [Feb. 22, 

the numbers of leucocytes present in the blood, and the degree of 
' bactericidal power ' possessed by the serum derived from it. 

The following is a short summary of our observations : 

(1) Methods employed. The microbe was grown in calves-foot 
bouillon for fourteen days. The culture fluid was then sterilized 
by heating in an autoclave to 115° C. for a quarter of an hour. 
Of the turbid liquid thus obtained \ to f cc. was injected into the 
lateral ear vein of a rabbit by means of a Koch's syringe. The 
temperature of the rabbit was taken in the usual way every half- 
hour, and when desired the animal was killed by cutting the 
carotid artery under antiseptic precautions. The blood was received 
into a sterilized centrifuge tube and was centrifugalised as soon 
as clotting had taken place. By this means a large quantity of 
serum can be obtained within a few minutes of the death of the 
animal. 

(2) The rise of temperature. Almost always such an injection 
was followed by a rise of temperature of 1 to 2| degrees Centigrade. 
A temperature of 41 , 4°C. was the highest that we have hitherto 
observed. The temperature generally rises quickly and within an 
hour after the injection may be two degrees above the normal. 
The maximum temperature is reached in 1^ to 2| hours after 
injection. The fall is far more gradual, but the temperature has 
generally returned to the normal within six hours of the injection. 

(3) Effect of the injection on the number of leucocytes present 
in the blood. Soon after the injection the number of leucocytes 
present falls to a fraction of the normal. Within an hour of the 
injection the larger leucocytes with lobed nuclei may have dimin- 
ished to so great a degree that several preparations have to be 
examined before one of them can be seen. The smaller white 
blood corpuscles however, consisting of one spherical deeply-staining 
nucleus and a trace of protoplasm (= lymphocyte), are not affected 
to so great an extent. Often scarcely any decrease in their 
number can be observed. At about the period of the acme of the 
fever a slight increase in the number of leucocytes is generally 
to be seen, so that almost as many are present as in the blood 
of a normal rabbit. At a later stage, when the fever has nearly 
passed off, a great increase in the number of leucocytes is always 
met with. In the majority of cases this increase is quite sudden. 
In one case for instance a preparation of blood taken 4|- hours 
after injection showed scarcely any of the larger leucocytes. A 
similar preparation made 4§ hours after injection, showed a well- 
marked leucocytosis. Five hours after the injection so many 
leucocytes were present that 40 to 50 could often be counted 
under the microscope in a single field of view. The increase affects, 
for the most part, the larger leucocytes with lobed nuclei. As a 



1892.] by the Injection of sterilized Vibrio Metschnikovi. 313 

rule only a slight increase in the number of lymphocytes present 
can be observed. After two days the blood still exhibits the same 
increased number of leucocytes. At the end of a week a certain 
amount of leucocytosis is still present, though the excess appears, 
in some of the few cases examined at this period, to consist to a 
great extent of the smaller lymphocytes. 

(4) Changes in the bactericidal power of the serum. Blood 
taken from a rabbit during the diminution stage of the number of 
leucocytes present, yields serum having less power of killing the 
Vibrio Metschnikovi than is normal. Blood taken as soon as 
leucocytosis has appeared yields serum which exerts a bactericidal 
action on this microbe which may be two or three times as great 
as that possessed by serum from an ordinary rabbit. It is note- 
worthy that this increase is not proportional to the increase in the 
number of leucocytes present. Blood taken twenty -four to forty- 
eight hours after injection yields serum which possesses a bacteri- 
cidal power for Vibrio Metschnikovi far above the normal. In one 
experiment a given volume of serum was found to be able to kill 
about 60 times as many microbes as could be killed by the same 
quantity of normal serum. 

(4) Note on the Method of Fertilisation in Ixora. By J. C. 
Willis, BA., "Frank Smart" Student in Botany, Gonville and 
Caius College. 

In Ixora salicifolia, D.C, the flowers are massed together in 
large corymbs, and thus rendered very conspicuous. A ring-shaped 
nectary upon the epigynous disc secretes honey, which is protected 
from rain and from short-lipped insects by the corolla-tube, whose 
length is 30 mm. and diameter about 1*5 mm. The anthers 
dehisce in the bud, introrsely, shedding their pollen upon the 
style, whose stigmas are tightly closed together, and thus protected 
from the pollen. When the flower opens, the stamens bend 
outwards and downwards until the anthers are below the rim of 
the corolla, when they usually fall off altogether. The style 
presents the pollen to insects visiting the flower. After the lapse 
of about twenty-four hours the stigmas separate from one another, 
and the flower is now in its second stage. The stigmas do not 
roll back so far as to ehect autogamy. 

A similar mechanism appears to occur in /. coccinea, and, so 
far as can be judged from dried specimens, in many other species 
of the srenus. 



314 Prof, Thomson, Experiments on Electric Discharge. [Mar. 7, 

Monday, March 7, 1892. 
Professor G. H. Darwin, President, in the Chair. 

The following communications were made to the Society : 

(1) Some Experiments on Electric Discharge. By Professor 
Thomson. 

A series of experiments were shown in which the electric 
discharge took place in bulbs without electrodes. It was shown 
that the colour of the discharge through the same gas varied 
very greatly with the density of the gas and the intensity of 
the discharge. This was illustrated by two bulbs each con- 
taining air; the discharge through one was a bright blue and 
through the other an apple-green. Another experiment showed 
that gas at a very low pressure could not act as an electromagnetic 
screen, though it did so at a high pressure. The laws governing 
the absorption of energy by conductors placed near very rapidly 
alternating currents were illustrated by experiments which showed 
that there was much greater absorption of energy by small pieces 
of tin-foil than large masses of brass or copper. 



(2) On the perturbation of a comet in the neighbourhood of a 
planet. By G. H. Darwin, F.R.S., Plumian Professor and Fellow 
of Trinity College. 

In Chapter II. of Book ix. of the Mecanique Celeste, Laplace 
considers the transformation of the orbit of a comet when it 
passes a large planet. His object is to show that the action of 
Jupiter suffices to account for the disappearance of Lexell's comet 
after 1779. 

He remarks that if a comet passes very near to Jupiter, it 
will throughout a small portion of its orbit move round the 
planet almost as though it were unperturbed by the Sun, and 
that both before its approach to and after its recession from 
the planet it will move round the Sun almost as though it were 
unperturbed by the planet. The nature of the orbit of the 
comet will usually be much transformed by its encounter with 
the planet. It is clear then that there must be some surface 
surrounding the planet which separates the region, inside of 
which the comet moves nearly round the planet, from the region 
in which it moves nearly round the Sun. Such a surface is to 
be found by the comparison of the ratio of the perturbing force 



1892.] Prof. Darwin, On the perturbation of a comet. 315 

to the central force in the motion round the Sun with its value 
in the motion round the planet. There is a certain surface at 
which this ratio will be the same in the two cases, and this is 
the surface required for the proposed approximate treatment of 
the problem. 

Now it does not appear to me that Laplace makes any attempt 
to show that such a surface is even approximately spherical, but 
he assumes that what has been called " the sphere of Jupiter's 
activity " is a true sphere, and determines its radius by the con- 
sideration of a special case. 

The object of the present note is then to treat this problem 
more fully than does Laplace, and to investigate the nature of 
the surface in question. 

It will appear that whilst Laplace's result is accurate enough 
for the purpose for which it is intended, yet a slightly different 
value for the radius of the sphere of activity would be more nearly 
correct. 

Let R, r be the radii vectores of Jupiter and of the comet, 
and let p be the distance of the comet from Jupiter. 

Let co be the angle between R and r, and 6 the angle between 
R produced and p. 

Let S, M, m be the masses of the Sun, Jupiter and the 
comet. 

Let P, T be the disturbing forces along and perpendicular 
to p, which act on the comet in its motion round the Sun ; let 
F be the resultant of P, T; and let C be the central force acting 
on the comet. 

Let ^, t£, Jf, (& be the similar things, also with reference 
to p, in the motion of the comet round Jupiter. 

Now we want to find a surface with reference to Jupiter 
such that outside of it the comet moves approximately in a conic 
section round the Sun, and inside of it in a conic section round 
Jupiter. 

If we consider a surface such that 

c or ' 

we shall have what is required. 

By the ordinary theory the disturbing function for the motion 
of the comet round the Sun as perturbed by Jupiter, is 

M \ ™ cos co 

\p R 2 

VOL. VII. PT. VI. 



316 Prof. Darwin, On the perturbation of a comet [Mar. 

But r cos eo = p cos + R, hence the disturbing function is 

irjI_I_4 C0S 



[p R R> 
Differentiating with respect to p and 0, we have 

P = -M \\ + —cos e 
[p M 

T=^sm0. 
Hence F 2 = M 2 jl + -1 + JL cos 1 

But C' = 5+JL\ 

r 

^72 J^-2 ^4 (■ 2 4 "J 

and thus _ = ^_ y!? |l + 2 ^co S « + t.|. 

Again the disturbing function for the motion of the comet 
round Jupiter as perturbed by the Sun is 

S \- +-775COS 
[r R" 

It will be seen that the sign of the second term is here + , 
because the angle between R and p is ir — 0. 

In this formula we have 

r 2 = p 2 +R 2 + 2pR cos 0. 
Plence differentiating with respect to p and 0, we have 

*® = S.Rfp-^\ sin 0. 

Now we might proceed to square these two and add them 
together to find jf 2 , and so go on to find the rigorous expression 
for Jp/CD, which equated to F'/G will give the rigorous equation 
to the required surface ; but the result would- be so complex as 
to be of little value because not easily intelligible. 

I therefore at once proceed to approximation. 



1892.] in the neighbourhood of a planet. 317 

S being very large compared with M and m, p will be small 
compared with r. 

Then since r = R 2 + p 2 + 2Rp cos 0, 

-K 3 -. o P a 3 P 2 , 15 P 2 ia 

Therefore approximately 

^=-l4l(i-l- 



3p - 3 p 2 15 p 2 j _ 

"i cos ^~2fe + Ti 2COS * lcos 



1 _ 3 C os 2 0_?^cos0+y^ cos 3 e\ 



and 



l 2 - + ^- 2 |(l-3cosW 



+ 3 cos (5 cos 2 $ - 3) (1 - 3 cos 2 0) -gj 



Asfain 



and 

^ 2 = + ^ 2 {9cos 2 ^(l-cos 2 ^) 

- 9 cos (5 cos 2 - 1) (1 - cos 2 0) £j , 
whence 

^ 2 = ^{ 1 + :1C0S ^- 12c0sS ^}- 
Now 



CD 

P" 
And 



) 2 ^{l + 3cos^-12cos^|} 



QL 2 \M+.m 

We now have to introduce a similar approximation into the 
value of F 2 /C 2 . 

25—2 



318 Prof. Darwin, On the perturbation of a comet [Mar. 7, 

4 

We have ^- 4 = 1 + 4 -^ cos 0, 



and therefore ^ = (-^-J ? (l + 4 £ cos *) 



F 2 _{ M V^Vi , a P 
P 
Equating F*/C* to JP/®*, we get 
'SVfS + m 



1 + 3 cos 2 5 



\p) ~[m) \M+mj 

- 4 cos (1 + 6 cos 2 6>) -£J 



: 8)*££>*ci + w«* 



_2 cos (1 + 6 cos 2 0) p 
5 " 1 + 3 cos' R 



Thus the equation to the surface is approximately 
R_ /S\ifS+m 
P 



-Of tiffin-"* 



2(M\i {3f+m\i cos (1 + 6 cos 2 0)\ 



5\8J \S + mJ (1 + 3 cos 2 0)™ J ' 

It is usually the case that m is negligible compared with M, 
and that M is also small compared with S, and in this case we 
may write the equation with sufficient accuracy 

!-(J)*<i+w«*. 

Laplace gives a formula for the radius of the sphere of activity 
which is virtually derivable from the above investigation on the 
special hypothesis that the three bodies lie in a straight line. 
Thus he puts equal to zero or 180° and finds, 



R .i 

— — 4ttt 


C4> 


p 


\M) 



But to find the true mean value of (1 + 3 cos 2 0) TT , we must 
estimate it all over the sphere. 

Now 

~ |T(1 + 3 cos 2 ey& sin 0d0dj> = T (1 + 3^ 2 )" dx. 

This integral evaluated by quadratures, is found to be equal 
to 1-063. 



1892.] in the neighbourhood of a planet. 319 

Thus the true mean gives 

-=1-063(4 
P \M 

Laplace makes it 

R .j_ ,Sy_ 



The ratio of the least to the greatest value of p in the formula 
suggested in this note is 1"149, and Laplace takes the minimum 
value of p as the radius of his sphere. 

In the case of Jupiter, Laplace's formula gives p = *054 R, and 
my formula gives p = '0o8R. 

It follows that Laplace's conclusion is sufficiently accurate for 
the purpose for which it is intended. 

(3) The change of zero of Thermometers. By C. T. Heycock, 
M.A., King's College. 

The author described the result of experiments he had made 
in conjunction with Mr Neville to overcome the change in zero 
which thermometers undergo when heated for a long time. The 
method consisted in boiling the thermometer for eighteen days 
in baths of either mercury or sulphur, at the end of this time 
the zeros were found to be practically fixed unless they were 
exposed to higher temperatures than those of the substance in 
which they were boiled. The paper was illustrated by a curve 
showing that the change in zero was very rapid for the first 
few hours, amounting in a special case to 11° C. for 20 hours 
heating, but that afterwards the change became almost nil as 
the heating was continued. 

(4) The Elasticity of Cubic Crystals. By A. E. H. Love, 
M.A., St John's College. 

(5) Changes in the dimensions of Elastic Solids due to given 
systems of forces. By C. Chree, M.A., Fellow of King's College. 

[Abstract] 

This paper deduces from a general theorem due to Professor 
Betti expressions for the mean values of the strains and stresses 
in any homogeneous elastic solid acted on by any given system of 
bodily and surface forces. Formulae for the mean strains in 
isotropic solids acted on only by surface forces were given by 



320 Mr Chree, Changes in the dimensions of [Mar. 7, 

Professor Betti, but he does not seem to have considered the 
general case, nor to have made applications such as those treated 
here. From the formulae for the mean strains the change can be 
found in the mean length, taken over the cross section, of any 
right cylinder or prism subjected to any given system of forces. 
Similarly the change in the whole volume of any elastic solid of 
any shape can always be expressed as the sum of a volume and a 
surface integral involving only the applied forces and the elastic 
constants of the material. 

Thus in an isotropic solid acted on by bodily forces whose 
components are X, Y, Z per unit volume, and by surface forces 
whose components are F, G, H per unit surface, the change Bv in 
the volume is given by 

SkBv = fJJ(Xcc + Yy + Zz) dxdydz + JJ(Fx + Gy + Hz) dS, 

where k denotes the bulk modulus, or m—^n in Thomson and 
Tait's notation. It is obvious from the equations of statical equi- 
librium that the position of the origin in the above expressions is 
immaterial. In any homogeneous aeolotropic solid the change in 
volume may be similarly determined, but the expressions under 
the integral signs are a little longer. 

The several formulae both for isotropic and aeolotropic solids 
are applied to a variety of special cases, a few of which will serve 
for illustration. The material to which the following results apply 
is, unless otherwise stated, assumed isotropic. 

When a solid of any shape is suspended from a point, or a 
series of points in one horizontal plane, its volume v is greater 
than if "gravity" did not act, and the increment Bv due to 
'•'gravity", represented by g, is given by 

Bv/v = gphjSJc, 
where p is the density and h the distance of the centre of gravity 
below the point, or points, of suspension. On the other hand, if a 
body be supported on a smooth plane, or at a series of points in a 
horizontal plane, its volume is diminished owing to the action of 
gravity, the diminution (— Bv) being given by 

-Bvjv=gph'/3Ic, 
where h' is the height of the centre of gravity above the plane of 
support. 

When a right cylinder or prism is suspended with its axis 
vertical its length I is increased, and the mean increment Bl taken 
over its cross section is given by 

8l/l=gpl/2E, 
where E is Young's modulus. When the cylinder rests on a 
smooth horizontal plane with its axis vertical, it shortens under 



1892.] Elastic Solids due to given systems of forces. 321 

gravity by an amount equal to the above. When the cylinder is 
suspended with its axis horizontal, in such a way that bending 
does not occur, it shortens, while when supported on a smooth 
horizontal plane in that position it lengthens. The alterations in 
the mean length in the two cases are given by 

Sl/l = + r/gph/E, 

where rj is Poisson's ratio, while h is the distance of the centre of 
gravity from the horizontal plane through the points of suspension 
in the first case and through the points of support in the second. 

When a solid of any shape rotates with uniform angular 
velocity eo about a principal axis of inertia through its centre of 
gravity the volume v is increased, the increment being given by 

Bv = co 2 I/3k, 

where i" is the moment of inertia of the body about the axis of 
rotation. 

W T hen a right cylinder or prism rotates about its axis it 
shortens, and the mean shortening (— SI) taken over the cross 
section is given by 

- Bl/l = 7JG) V7^> 

where k is the radius of gyration of the cross section about the 
axis. 

When a rectangular parallelepiped 2a x 2b x 2c rotates about 
the axis 2c, the mean increment 28a in the distance between the 
faces perpendicular to 2a is given by 

8a/a = co 2 p (a 2 - V b 2 )/2E. 

Thus the tendency to increase in length in material lines 
perpendicular to the axis of rotation becomes reversed when the 
dimension perpendicular to this and to the axis of rotation is 
sufficiently great. 

A homogeneous sphere, whether isotropic or aeolotropic, owing 
to the mutual gravitation of its parts suffers a diminution in 
volume given by 

— Sv/v = gpRjbk, 

where R is the radius and g "gravity" at the surface. This 
suffices to prove that the application of the mathematical theory 
of elasticity to the earth, treated as a homogeneous solid, violates 
the fundamental condition that the strains must be small, unless 
the material be assumed to offer a much greater resistance to 
compression than any known material under normal conditions at 
the earth's surface. 

The change in volume due to the mutual gravitation in its 
parts in any very nearly spherical bod} 7 , when isotropic, is shown 



322 Mr CJtree, Changes in the dimensions of Elastic Solids. [Mar. 7, 

to be the same as in a sphere of the same material of equal 
volume, and it is thence concluded that the spherical is a form in 
which the reduction of volume due to gravitation is in general 
either a maximum or a minimum. The reduction of volume is 
calculated for a gravitating ellipsoid, and it appears that the 
sphere is the form in which, when the volume is given, the re- 
duction is a maximum. In a nearly spherical ellipsoid whose 
principal sections through the longest axis are of eccentricities e x 
and e 2 the reduction in volume is given by 

- Bvfv = (gpR/rok) {1 - (e* - e t V + e/)/45}, 

where R is the radius, and g the value of gravity at the surface, 
in a sphere of equal volume and density. 

In a given volume of an aeolotropic material a very slight 
assumption of an ellipsoidal form, insufficient to produce an 
appreciable effect if the material were isotropic, increases or 
diminishes the diminution in volume due to mutual gravitation 
according as it consists in a lengthening or a shortening of those 
material lines which are parallel to directions in which the 
linear contraction under uniform normal pressure is above the 
average. 



(6) On the law of distribution of velocities in a system of 
moving molecules. By A. H. Leahy, M.A., Pembroke College. 

1. The following proof appears briefly to establish the fact 
that Maxwell's law of distribution of velocities gives the only steady 
distribution. The proof is a little shorter than the ordinary proof 
as given by Boltzmann, even if Mr Burbury's variation of it as 
published in the Philosophical Magazine for October 1890 be 
adopted. 

Let a particle whose velocity is OP in magnitude and direction 
strike a particle whose velocity is op. Suppose the particles to 
belong to different systems, and let the number of particles of 
the first kind which have velocity components lying between £ 
and g+d];, w and w + dv, £ and £+d£, where £, n, £are the com- 
ponents of OP, be F(0P)d^dnd^. Let f {op) d%dw' d£ have a 
similar meaning when applied to the particles of the second 
system. Then the number of impacts which particles with ve- 
locity OP have with particles of the second system which have 
velocity op will, in the interval dt, be per unit volume 

F(OP)d^d v d^.7rs' 2 udt.f(op)d^d v 'd^ (1), 

since each particle in the unit volume strikes, on the average, 
7rs 2 udtf{op)d^'dn'd^' in the interval dt; the particles bejng re- 
garded as hard spheres, the sum of the radii of two spnefes, one 



1892.] Mr Leahy, On the law of distribution of velocities. 323 

of each system, being s, and u the velocity of a particle of the 
first system relative to the velocity of a particle of the second 
system. 

Suppose now that after encounter the velocities of the particles 
become OP i , op t respectively, so that the components of 0P 1 lie 
between £ and £ + (*£, V l and rj^drj^ £ and £ + «&£; and f/, 
77/, £/ have similar meanings as the components of op v Since 
the velocity of the centre of gravity is unchanged by the impact, 
the condition that the required change shall take place is that the 
direction of u shall lie within a cone of solid angle dS making an 
angle 6 with the line of centres at impact. A collision such that 
velocities OP, op before collision may become velocities 0P 1} op t 
after collision may be called a " collision of a given kind," and 
since, as Mr Burbury has pointed out, all directions of the relative 
velocity after encounter are equally probable if the molecules 
behave as hard elastic spheres, the whole number of encounters of 
the given kind in the interval dt will be 

JO 

F(OP)f(op)d^d V d^.d^d v 'd^.irshidt.^ (2) 

per unit of volume. 

Conceive now that the velocity of every molecule of the system 
is suddenly reversed in sign, the molecules of the solid boundary 
of the system, if such exist, being similarly reversed as to the 
direction of their velocities, the position of every molecule being 
unaltered. The physical properties of such a medium will of 
course differ in many respects from those of the original medium, 
but it will at any rate have this property, that all particles whose 
velocities in the original medium changed from OP to 0P l in the 
time dt will in the second (or as we may call it the "reversed") 
medium change in a second interval dt from — 0P 1 to — OP. 
Now, since the distribution of particles is perfectly regular in 
space, the distribution of velocities in the original medium 
" behind " any class of molecules is the same as the distribution 
" in front " of the same class. Hence the number which in the 
reversed medium change their velocity — 0P t for — OP by striking- 
particles with velocity op t is in the interval dt 

d ST 
F{- OPJfi- o Pl ) %&& ■ m&nlMU ■ -rrshidt . — . . .(3), 

where u' is the velocity of 0P X measured relatively to op 1 and dS' 
is the angle of the cone, whose axis makes an angle 6 with the 
line of centres, within which the direction of the relative velocity 
u' must lie in order that the collision may be one of the given 
kind. Since the number which change from velocities OP to 
velocities 0P 1 in the original medium by collisions of the given 



321 Mr Leahy, On the law of distribution of velocities [Mar. 7, 

kind is equal to the number which change from 0P 1 to — OP in 
the reversed medium by collisions of the same kind, expressions 
(2) and (3) are equal. Also u' in (3) is the same as u in (2) since 
the velocity of the centre of mass is unchanged and the spheres 
are elastic, and by ordinary geometry dS' is equal to dS and 
d£dr)d£ . d%'dr)'d£' is equal to d^drj^^.d^dr/^d^. Hence, since 
F(—0P^) in the reversed medium is equal to F(OP^) in the 
original one, 

F(OP 1 )f(o Pl ) = F(OP)f(op), 

and therefore since the kinetic energy is unchanged by the impact 
we have, by the ordinary methods, Maxwell's law of distribution 
namely 

OP* 

F(OP) = Ae 

In order to examine the validity of the above proof the 
assumptions underlying equations (1) and (2) should be further 
considered. In equation (1) the assumption is that, if a number 
of particles are distributed uniformly throughout a medium, 
and if the velocities are so distributed that the number per 
unit volume which have velocity op is f(op) d% 'drj 'dg ', then the 
number in the volume F(0P)d^drjd^.7rs\(,dt, which may be 
written do v is f (op) d^'drj'd^' . do v This assumption is equivalent 
to two others, first, that the particles are distributed throughout 
the medium with perfect uniformity so that we can safely take 
the number of molecules of a given kind in an element of volume 
to be proportional to the volume of the element if the element is 
large enough to contain a great many molecules ; secondly, that 
the particular volume do x is large enough for the first assumption 
to be applied to it. In order to prove result (2) we must suppose 
that the number of molecules within the volume 

j a 
F(OP) d^drjd^. irs'udt . ^cos 6, 

which may be written do 2 , is proportional to do r Thus, since the 
volume do t is greater than do 2 , the whole assumption that we 
make is that do 2 , and consequently do 1 , is large enough to contain 
a very large number of the molecules which are distributed so that 
the number which have a velocity op is given by the function 

/(op)- 

To estimate the number of these molecules, suppose F(0P) 

N 0F * 
to be equal to — — e * 2 , which is Maxwell's law of distribution. 

7r 2 a 3 
Then, taking hydrogen as an example, since* Nits' 1 is 7"0 x 10 3 

* These numbers are calculated in accordance with Professor Tait's results, 
Edin. Trans, xsxin. p. 91. 



1892.] in a system of moving molecules. 325 

approximately, the volume is 

7-0 x 10 3 -9^1 dPd V d£ u dS 

= e « 2 . ' . udt . -. — cos 6 

,,1 a 3 4>7r 

cubic millimetres. Now a cubic millimetre of hydrogen at atmo- 
spheric pressure contains about 976 x 10 16 molecules. Hence the 
number of molecules in do 2 is 

. n ,-„. -9H dPd-ndt 1± dS n 
6-8 x 10 20 .e a« . , * . udt . -r— cos 0. 
a 3 47r 

Suppose u to be equal to k times a which is in hydrogen equal to 
7"62 x 10 5 millimetres per second; we get the whole number of 
molecules in do 2 to be 

K _ -911 dPdr/dt ,-„„, .dS n 

5-2 . e a» . , . W 6 hdt -j- . cos 0, 
a 47T 

tZi being measured in seconds ; and this number must be large 
in order that equation (2) may accurately give the number of col- 
lisions of the given kind. 

The smallest value which we can ascribe to dt will depend 
upon the magnitude of the limits d%, drj, d£, dS which define the 
encounter. Suppose that d% — drj = dt, = a/1000 ; suppose also that 
dS/4<Tr = 10" 3 . Let us also suppose OP not to be greater than 
2a, a supposition which excludes less than 0'5 per cent, of the 
whole number of molecules. Suppose also that k is greater than 
10~ 2 , so that the relative velocity u is not less than a/100. These 
suppositions give the whole number of molecules in do 2 to be 
greater than 9"6 dt . 10 10 , so that this number is more than a 
million if dt is not less than 10~ 5 of a second. This estimate of 
the limiting value of dt is perhaps too small as we have taken 
the limits of d^drjdt, exceedingly small, but it will appear that 
dt must not be taken indefinitely small and should at any rate 
be greater than the mean time between collisions, which is of the 
order 10~ 9 of a second. With the above proviso as to the value 
of dt, it appears that result (2) can be taken to be accurate, and 
since result (3) is merely an application of result (2) to the re- 
versed medium it appears that the assumptions made can be 
relied on. 

2. It has throughout been assumed that the distribution of 
velocities is " steady ", and the proof shows that Maxwell's law 
gives the only possible steady distribution. It is however desirable 
to show if possible that the system must ultimately acquire a 
steady distribution. Now the steadiness of the distribution has 
been assumed twice, first when the assertion is made that the 



326 Mr Leahy, On the law of distribution of velocities [Mar. 7, 

number of molecules with velocity op struck by molecules moving 
among them with relative velocity u is proportional to udt.nrs*, 
secondly, when the distribution in the "reversed" medium was 
taken to be the same as that in the original one. If the distri- 
bution is not steady, expression (2) must be amended by inserting 
a factor (1 + vdt), and expression (2) must contain a factor 
(l + v'dt); where v, v depend upon the variations of F and f. 
The result obtained as before will be that 

F(OP 1 )f(o Pl )-F(OP)f(op) 

will not be zero but equal to wdt where w depends upon v, v', F, 
and f. Integrating equation (2) and using the proposition that 
all directions of encounter are equally probable, we get the usual 
result 

j t F(OP) = F(OP) d£d v dzf[Jd%'d V 'd?7rs 2 u 

i- \j\(F{OP\)f{o Pl ) - F(OP) f{op)) ds} , 

where dt~, drj', d£' are elementary increments of the components 
of op, and the double integral is taken for all possible impacts 
between particles whose velocities before collision were OP, op; 
OP v op t being the velocities which the particles acquire if their 
relative velocity falls within the cone of solid angle dS. 

Since the subject of integration in the double integral is equal 

d 
to wdt, j,F(OP) must contain dt as a factor and will be very 

small when dt is very small. But, since there is a limit to the 

d 
minimum value of dt, this does not prove -y-F{OP) to be zero; 

(Lb 

that is we cannot in this way prove the ultimate distribution to 
be steady, although its variation from the steady state must be 
small when the distribution of the particles is regular throughout 
the space considered. 

Boltzmann's proof would show that the function H which he 
has introduced will continually diminish until the steady state is 
obtained, but I think that it assumes equations (1) and (2) to be 
absolutely true, which they appear to be if the motion is from 
the first assumed to be steady. The proof that the motion of the 
particles finally must attain a steady state is apparently still 
wanting, although the above argument shows that the divergence 
from the steady state must ultimately be small. It is not im- 
possible that F(OP) may ultimately be periodic with a period of 
magnitude of the same order of magnitude as the time of free 
path. But the assumption that the motion of the particles is 



1892.] in a system of moving molecules. 327 

ultimately absolutely steady is after all not greater than the 
assumption that it is ultimately perfectly regular, and if the 
regularity of the distribution both in space and time is assumed 
Maxwell's law of distribution appears, from the above, readily to 
follow. 



Monday, May 2, 1892. 
Prof. G. H. Darwin, President, in the Chair. 

The following were elected Fellows of the Society : 

Thomas Clifford Allbutt, M.D., F.R.S., Fellow of Cams College, 

Regius Professor of Physic. 
David Sharp, M.A. (M.B. Edin.), F.R.S., Curator in Zoology. 
J. C. Willis, B.A., Gonville and Caius College. 

The following was elected an Associate : 

A. Antunis Kanthack (M.B. Lond.), St John's College, John 
Lucas Walker Student in Pathology. 

The following communications were made to the Society : 

(1) The application of the Spherometer to Surfaces which are 
not Spherical. By J. Larmor, M.A., St John's College. 

The ordinary form of spherometer, which is used for measuring 
the curvatures of lenses, rests on the surface to be measured by 
three legs which are at the corners of an equilateral triangle ; and 
the mode of using it consists in finding the length of the ordinate 
drawn up to the surface from the centre of the triangle formed by 
the points of support, by means of a micrometer screw moving 
along the axis of the instrument. 

In the actual use of the instrument the surface to be measured 
is assumed to be spherical ; and the question has apparently not 
occurred to examine the character of the results which may be 
derived from its application to a surface of double curvature. 

On actual trial with such a surface, for example the cylindrical 
surface of an iron pipe, it appears at once that when the centre is 
set at a given point the instrument may be rotated anyhow on its 
axis without affecting its reading. It therefore measures some 
definite quality of the double curvature of the surface at the 
point. There is a temptation to hastily assume that the plane of 
support is parallel to the tangent plane at the centre of the instru- 



328 Mr Larmor, The application of the Spherometer [May 2, 

ment, that it is in fact the indicatrix plane of that point, and to 
deduce that the reading gives the mean of the principal curva- 
tures of the surface; this result is correct, but the assumption just 
mentioned is erroneous. 

To obtain a rigorous investigation, let us assume that the 
points of support form an isosceles triangle, let the base subtend 
an angle 2a at the centre of the circumscribing circle, and let c be 
the radius of this circle and h the ordinate drawn from its centre 
up to the surface. If this ordinate is taken as axis of z, the equa- 
tion of the surface will be 

,2 2 

where (p, q, — 1) is the direction of the tangent plane at the origin, 
and R , R 2 are the radii of principal curvature. As the three legs 
rest on the surface, we have 

7 //i x • //i x 1 2 fcos 2 (0 + a) sm 2 (0 + a)) 
h = cp cos (0 + a) + cq sin (0 + a) + \<? \ K R ' + ^ '-I 

7 //i \ • //i x t o(cos 2 (# — a) sin 2 (0 — a)} 

h = cp cos (0 - a) + cq sin (0 - a) + |c 2 ^ H ^ -\ , 

[ K 1 K 2 ) 

7 n -a .12 f cos2 & sin 2 
h = — cp cos — cqsm0 + Jc <— ~ 1 — ^ — 

where it + is the azimuth of the vertex of the triangle of sup- 
port. We are to eliminate p, q, and so connect h with R v R 2 and 
0. By addition of the^ first pair of relations 

2h = 2cp cos cos a + 2cq sin cos a + 2 c ) ( p~ + 75- ) 

+ (V - »-) cos 2 # cos 2a 
therefore by use of the third 
2h (1 + cos a) = -k 2 i(^ + Jj-) (1 + cos a) 

+ (jt- - -=-} cos 20 (cos 2« + cos a)l , 
or on rejecting the factor 1 -f cos a, 

7 = (i + i) + (1; - s-j cos 2 ^ 2 cos a - !)• 

The value of h therefore depends on the azimuth except in 
one case, when a is ^ir so that the triangle of support is equi- 
lateral, which is the case referred to above. The quantity involved 



18.92.] to Surfaces which are not Spherical. 329 

in the formula is then -rr+ t> ; and by referring back to the 

original case of a spherical surface we see that the instrument 
measures the arithmetic mean of the principal curvatures. 

Thus for example the equilateral form of the instrument may 
be conveniently used to measure the curvature of a cylindrical 
lens or a cylindrical pipe, but for that purpose its indication must 
be doubled. 

The equilateral form will be of no use for testing deviation 
from sphericity at a given point of a surface. The isosceles form 
may however be so used, the difference of the extreme curvature- 
indications given by it for any point being by the above formula 



(srs:) (2cosa ~ 1) ' 



that is directly proportional to the difference of the principal 
curvatures. The curvature may thus be completely explored*. 

In all these formulae the usual assumption is made that the 
span of the instrument is small compared with the radii of curva- 
ture of the surface. 

If the instrument had four legs at the corners of a rectangle, 
there would be only two positions in azimuth, corresponding to the 
sections of greatest and least curvature, in which it would rest 
firmly at a given point on a surface, with all its legs in contact ; 
and the plane of contact would in this case be parallel to the 
tangent plane at the summit of the surface. The readings for 
these positions would give 

cos 2 a sin 2 a. 

. sin 2 a cos 2 a 

and ~R~ + ~~R~ ' 

where a is an angle made by a diagonal of the rectangle with a 
side ; so that the values of both principal curvatures might thus 
be determined. 

* Mr H. F. Newall informs me that an isosceles spherometer is used by 
Dr Common for exploring the curvatures of his large specula. 



330 Prof. Thomson, On the electric strength of a gas. [May 1G, 

May 16, 1892. 
Professor G. H. Darwin, President, in the Chair. 

The following were elected Fellows of the Society : 

W. Robertson Smith, M.A., Fellow of Christ's College, Professor 
of Arabic. 

J. K. Murphy, B.A., Caius College. 

The following communications were made to the Society : 

(1) Recent advances in Astronomy with Photographic Illustra- 
tions. By H. F. Newall, M.A., Trinity College. 

A series of photographs was exhibited by the lantern and 
described, to illustrate recent progress in astronomical photography. 
The series included some interesting specimens taken with the 
Newall telescope, in which the object glass is not specially cor- 
rected for photographic purposes. 

(2) On the pressure at which the electric strength of a gas is a 
minimum. By J. J. Thomson, M.A., Cavendish Professor. 

The author showed that when no electrodes are present, the 
discharge passes through air at a pressure somewhat less than 
that due to 1/250 mm. of mercury ; the discharge passes with 
greater ease than it does at either a higher or a lower pressure. 
Mr Peace has lately shown that when electrodes are used, the 
critical pressure may be as high as that due to 250 mm. of mercury : 
so that as the spark length is altered the critical pressure may 
range from 250 mm. to 1/250 of a mm. It was pointed out that 
this involved the possession by a gas conveying the discharge of a 
structure much coarser than any recognized by the Kinetic Theory 
of Gases. The author suggested a theory of such a structure and 
showed that the theory would account for the influence of spark 
length and pressure on the potential difference required to produce 
discharge. 

(3) On a compound magnetometer for testing the magnetic pro- 
perties of iron and steel. By G. F. C. Searle, M. A., Peterhouse. 

When a bar of iron or steel is subjected to the action of a 
longitudinal magnetic force, H, it is found that the intensity of 
magnetisation of the iron thereby produced depends not only 
upon the value of H at the instant, but also upon the series of 



1892.] Mr Searle, On a compound magnetometer. 



331 



values which H has previously assumed. Thus if the magnetising 
force is made to undergo a series of changes in a cyclical manner, 
the curve representing the relation of / to H will be of the form 
of a loop, which however degenerates into a straight line when 
the maximum value of H does not exceed 04c.G.s. units*. This 
dependence of the intensity of magnetisation, due to a given 
magnetic force, upon the previous magnetic history of the iron 
has been called by Ewing hysteresis. The curve ABCDEFA (fig. 1) 




Hysteresis Curve for Annealed Steel Wire. 



taken from Prof. Ewing's book on " Magnetic Induction in Iron 
and Other Metals " will serve to give a general idea of the relation 
between / and H for a piece of annealed pianoforte steel wire 
when the magnetic force is made to pass repeatedly through a 
complete cycle of changes. The maximum values of H and / in 
this curve are about 100 and 1100 C.G.S. units respectively. The 
curve OB gives the relation between / and H for a piece of steel 
which has never previously been magnetised or which has been 
completely demagnetised by continued reversals of a magnetic 
%force whose amplitude has been slowly diminished to zero. 

These hysteresis curves are of great interest from the practical 
as well as from the philosophical point of view, since, as has been 
shown by Warburg and by Ewing-f-, the area of the curve, when 
estimated on the proper scale, represents the energy expended 
per cubic centimetre of the iron in carrying H through its cycle 
of changes. 

In determining the form of the hysteresis curve, a specimen 
of the material in the form of a wire is placed inside a long 
uniformly wound solenoid through which a current can be sent. 
The current gives rise, inside the solenoid, to a uniform longi- 
tudinal magnetic force whose value can be calculated from the 



equation 



H = 4fTTlU, 



* Lord Eayleigh, Phil. Mar,. March, 1887. 

t J. A. Ewing, "Magnetic Induction in Iron and Other Metals," § 79. 

VOL. VII. PT. VI. 2G 



332 Mr Searle, On a compound magnetometer for [May 16, 

where i is the strength of the current, and n the number of turns 
of wire upon the solenoid per centimetre of its length. The 
solenoid is placed near a suitable mirror magnetometer; and a 
small coil, which is joined up in series with the magnetising sole- 
noid, is so adjusted that it exactly neutralises the action of the 
solenoid itself upon the magnetometer. Thus when the specimen 
of iron is placed inside the solenoid the deflection produced is 
due entirely to the magnetisation of the specimen. From the 
observed deflection the value of the intensity of magnetisation, /, 
can be determined. • The magnetising current also passes round a 
suitable galvanometer by means of which its strength, i, can be 
measured. The strength of the current is gradually varied by 
means of a resistance box in the circuit, and the simultaneous 
readings of the galvanometer and magnetometer are noted. The 
values of H and I deduced from these readings are used as abscissa 
and ordinate in the construction of the hysteresis curve. This 
process naturally involves a good deal of labour. 

I have endeavoured to construct an instrument which should 
perform simultaneously the functions of both galvanometer and 
magnetometer and should cause a spot of light -to trace out -a 
rrysteresis curve upon a screen. One method of attaining this 
end is to provide a mirror with two independent motions about 
two axes mutually at right angles, the motions about these two 
axes being governed by two small magnets. One of these magnets 
must be acted on by a magnetic force proportional to the 
magnetising current, and the other by a magnetic force propor- 
tional to the intensity of magnetisation of the specimen. 




Fig. 2. 



This idea was put into practice in the following manner. AB 
(fig. 2) is a thin aluminium wire about 80 centimetres long. This 
is suspended by one end A by a silk fibre from the support K. 



1892.] testing the magnetic properties of iron and steel. 333 



Near its top the wire carries a small magnet G whose axis is at 
right angles to the wire. The lower end B of the wire carries a 
small fork BBE, also of aluminium wire, across which the silk 
fibre BE is stretched. Attached to this fibre by means of wax is 
a small plane mirror F, such as is used in reflecting galvanometers, 
carrying a small magnet whose axis is at right angles to the fibre 
BE. Attached to the bottom edge of the mirror is a disk of 
thin mica about 1 inch in diameter. When the plane of the 
mirror is vertical, the plane of the disk is horizontal. Close 
beneath the mica disk is placed a piece of cardboard in a horizontal 
position. The mica disk, owing to the close proximity of the 
cardboard, very rapidly reduces the mirror to rest. The mirror is 
fixed to the fibre so that the centre of gravity of the mirror and 
mica disk is slightly below the line BE. Thus the controlling 
force acting on the lower system consists partly of gravity and 
partly of the magnetic force due to the earth and to any control 
magnets which may be required to bring the mirror into any 
desired position. The mirror now possesses two independent mo- 
tions, the one about the axis AK and the other about the axis BE. 
The apparatus is set up as in fig. 3*, in which the suspended 
part has been turned through a right angle so that the mirror is 



K 



n c 



Fig. 3. 




I 



now seen edgeways. The plane of the paper is supposed to be a 
plane through the wire AB at right angles to the magnetic 
meridian. The magnet G is therefore at right angles to the plane 
of the paper. The mirror F is shown slightly tilted. The coil L 

* This figure is purely diagraniatical and does not represent the relative pro- 
portions of the separate parts of the apparatus. 

26—2 



334 Mr Searle, On a compound magnetometer for [May 16, 

is placed near the magnet G and deflects it through an angle 
proportional to the strength of the current in L, the deflections 
being kept very small. The solenoid M is placed in a vertical 
position east or west of the mirror, its upper end being about in 
a horizontal line with the mirror. The small coil N can be 
adjusted so that the effect on the magnet F of the solenoid itself 
is completely neutralised. R is a resistance box for varying the 
current, P a battery, and Q a commutator. A lens S of about 40 
inches focal length forms the window of the case in which the 
suspended part is hung. A lamp and screen are placed at about 
40 inches from the lens. Cross wires are placed in front of the 
lamp and a sharp image of these is thrown upon the screen by 
the action of the lens and mirror. The spot of light may be 
made to take up any desired position on the screen by properly 
adjusting small permanent magnets in the neighbourhood of the 
two magnets G and F. I had expected that a good deal of trouble 
would have been caused by change of zero in the vertical direction 
owing to changes in the silk fibre on which the mirror is strung, 
but I was agreeably surprised to find that the spot of light would, 
if the mirror were disturbed, return to the same horizontal posi- 
tion to within -^ inch. The zero position seemed to be quite 
permanent. 

In order that the two motions of the spot of light should take 
place in horizontal and vertical lines, the axis DE must be ad- 
justed so as to be accurately perpendicular to the axis of suspension, 
AK. The necessary fine adjustment is easily made by slightly 
bending the suspending wire near the point B. I found that a 
small block of cork formed the best means of connecting the wire 
AB with the fork DBE. To get rid of any secondary effect of 
the coil L upon the lower magnet a second small " compensating " 
coil may be included in the circuit. In order to bring the spot of 
light quickly to rest a suitable mica vane was attached to the 
vertical wire AB. This rapidly stops the motions in azimuth. 

When I exhibited the instrument to the Society, the magnet 
G was slightly affected by the induced magnetization of the speci- 
men of iron in the solenoid. This effect can not be compensated 
by another coil, since a coil through which the magnetising current 
flows will not imitate the magnetic behaviour of the iron. To 
remedy this defect I have fitted a second magnet of moment 
nearly equal to that of C to the vertical wire a short distance 
below C, its axis pointing in the opposite direction to that of G. 
The effect of the magnetised specimen on the astatic system is 
very small and I hope that all trouble from this source has now 
been got rid of. 

The indications of the instrument can easily be reduced to 
absolute measure (at least approximately) in the following way. 



1802.] testing the magnetic properties of iron and steel. 335 

Suppose, for instance, that it is desired that a movement of the 
spot of light through 10 centimetres horizontally should corre- 
spond to a magnetic force iuside the solenoid equal to 100 C.G.S. 
units. A known current is sent round the coil L, and this coil is then 
so adjusted that the spot of light is deflected through the proper 
distance cor-responding to the calculated value of H. To standardise 
the vertical motion of the spot, a magnetised steel wire may be 
placed inside the solenoid M, through which no current is passing, 
and the solenoid is then adjusted until the spot of light shows a 
deflection in the vertical direction of 10 centimetres for each 1000 
C.G.S. units of intensity of magnetisation of the steel wire. The 
intensity of magnetisation of the steel wire can be determined by 
the use of an ordinary magnetometer. If the cross section of the 
wire to be tested is different from that of the steel wire an 
appropriate factor must be introduced. 

Although for very accurate observations in the subject of 
hysteresis the use of two separate instruments, galvanometer and 
magnetometer, will probably still be necessary, yet I think that 
the instrument I have described may be found useful as a means 
of rapidly gaining an approximate knowledge of the form of the 
hysteresis curves for various samples of iron and steel without 
any calculation. For this purpose it may be a useful instrument 
in the Lecture Room. 



May 30, 1892. 
Professor G. H. Darwin, President, in the Chair. 

The following communications were made : 

(1) The hypothesis of a liquid condition of the Earth's interior 
considered in connexion with Professor Dariuins tlieory of the 
genesis of the Moon. By Eev. O. Fisher, M.A., F.G.S., Hon. 
Fellow of Jesus College. 

In a series of papers in the Philosophical Transactions, Parts I. 
and II. 1879, Professor Darwin has developed the theory of tidal 
action in the solar system. 

At p. 23 of his paper on Bodily Tides of Spheroids he 
gives " a dynamical investigation of the effects of a tidal yielding 
of the earth on a tide of short period according to the canal 
theory." A numerical estimate will afford a clearer idea of the 
effects produced. 



336 Rev. 0. Fisher, On the hypothesis of a [May 30, 

The symbols used are 

h = the depth of the canal. 

a = the earth's radius. 

w = the rotational speed referred to the moon (23 h. 56 m. 
= lunar day at present). 

(j) — (ot = the longitude west of the moon. 

e = half the lag of the bodily tide. 

2E — the greatest range of the bodily tide at the equator*. 

3 

t = ~ x the moon's mass x a 2 -r- (her distance) 3 . 

The result obtained for the height of the wave relatively to 
the bottom of the canal is 



h - h d X > 



where 

C R 
dx 



- sw^r {(I cos e - 1 ° E ) cos 2 ( * " "*) 

+ | sin e sin 2(<j> — oit)if. 

Taking up the investigation from this point since the land 

partakes in the rise and fall of the bottom of the canal, the 

measurable height of the tide will be the difference between the 

depth of the canal and the height of the water above the bottom 

d£ 
of it, which will be — h ~- , or 

aW-gh 1(1 C ° S 6 ~ i 9 J C ° S 2 ^~ (ot ^ + l sin e sin 2 & ~ ^ } ' 

h 

For -^—5 r write H, for 2 (6 — cot) write 6. 

aW—gh vr / 

Then the height of the tide will be expressed by 
-#jcos(0-e)-g^cos-0 

= — ITJcos 6 (cos e — v — ) + sin#sineL 

* Elsewhere Prof. Darwin uses E as the ratio of the bodily tide in the case of 
viscosity to the like in the case of fluidity. 
+ Loc. cit. p. 26. 



1892.] liquid condition of the Earth's interior. 337 

We must now estimate the numerical value of ■= ■ — . 

5 r 

From the definition of t already given we have 

4 gE 4 2 (distance) 3 1 earth's mass r 

5 t 5 3 moon's mass a' 2 a 2 

_ _8_ earth fd\ 3 E 
15 moon \a/ « ' 

= 15 0-01228 ( 60 ' 2634 )' 90902404 ' 

= 045474.fi"; 

a foot being the unit. 

If the interior is considered liquid, the bodily tide may be 
taken equal to the equilibrium tide, which would be about If feet 
from highest to lowest - !", and E would be half that, so that 

!#£ = 0-39789. 

O T 

The lag of such a tide would be small. Darwin seems to con- 
sider 14' as an admissible value for 2e in that case, and cos e would 

be 0-99996C8. Hence cos e would be greater than - — , and we 

& o T 

may assume 

sin e 



4<r/E 
cos e — - - — 

O T 



= tan D, 



and by substitution and reduction we obtain for the height of the 
measurable tide 

- E |l - 2 cos £ x | ^ + (| ^)*l* cos {2 (</> - cot) - D), 

or, cos e being very nearly unity, 

-hU ~\^\ cos {2(0- cot)- B], 

= -Hx 0-60211 cos (2 (<f> - art) - £>}. 

Hence the tide would be fths of what it would be on a rigid 
earth. 

It is evident that tan D is small. Hence low water will occur 
a little west of the moon. 

* " The Moon " by Proctor, Tab. iv. p. 313. 
t Thomson and Tait, § 804, 2nd Ed. 



338 Rev. 0, Fisher, On the hypothesis of a [May 30, 

In forming the potential of the protuberance of the "bodily tide 
the earth has been taken as homogeneous. But the superficial 
parts having only half the mean density, it seems that the value 

& aft 1 

of - — ought to be taken at one half that assumed, and then 



^^?= 0-19894, 
5 T 



and we find for the tide 



- H x 0-80106 cos (2 (<£ -cot)-D 



which shows that it will be diminished by only ^th of what its 
height would be if the earth was rigid. 

We learn from this expression that high ocean tide will occur 

7T D 

when 2 (<£ — cot) — D = ir, that is when <£ — cot = -= + -~ to the 
west of the moon ; and high earth tide will occur when <£ — cot = — -= , 

or - to the east of the moon. Hence the crests of the ocean and 

2 

earth tides are separated by the obtuse angle -= H = — , so that 

the tidal protuberances of both of them, which are nearest to the 
moon, are to the east of it ; and the effects of the couples caused 
by the moon's attraction upon both of them will be to retard the 
earth's rotation. 

Prof. Darwin remarks, that the expression for the height of the 
ocean tide as affected by the bodily tide is subject to a modifica- 
tion of the same form on the equilibrium theory as on the canal 
theory, with the exception of a change of sign. Hence on that 
theory also, which neglects the inertia of the water, and therefore 
less nearly represents the case of nature, the ocean tide would be 
diminished by the same factor, and therefore only to the small 
extent of about one-fifth, as has been now shown would be the 
case on the canal theory. 

The lag of the bodily tide has here been put at 14', because 
Darwin has shown* that, on the hypothesis of approximate 
liquidity, the reaction of the moon on the bodily protuberance 
with that amount of lag would account for the unexplained 
acceleration of the moon's mean motion at the present time of 
4 seconds in a century. It is evident that if the lag of the bodily 
tide is larger, cos 2e will be smaller, and the reduction of the tide 
in the canal will be still less. 

' * "Precession of a viscous spheroid,'' § 1-4. 



1892.] liquid condition of the Earth's interior. 339 

The above appears to be a sufficient answer to the objection 
brought against the theory of internal liquidity that in such a 
case there could be no measurable ocean tides. 

Prof. Darwin appears when he wrote to have held the view 
that the earth must be very rigid probably in consequence of his 
investigation by which he had proved that on a solid globe 
nothing short of a high degree of rigidity could sustain the 
weight of continents and mountains. This necessity is of course 
entirely removed by Airy's hypothesis that the crust is supported 
in a state of approximate hydrostatic equilibrium on a yielding 
nucleus*. 

Assuming therefore the necessity of a high degree of rigidity, 
Darwin finds a certain coefficient of viscosity, which according to 
his calculations would cause the obliquity of the ecliptic to 
increase most rapidly at the present time (p. 526), and uses this 
particular value in his numerical calculations. Thus, when he 
estimates the length of time, since the moon may have been 
detached from the earth, at about 57 million years -J*, the estimate 
depends upon that particular value of the viscosity. So also do 
his estimates of the amounts of heat generated by tidal friction 
within the earth during certain intervals of time dating from the 
same epoch. And in short all the numerical results in Table IV. 
at p. 494, depend upon the particular assumed high degree of 
viscosity. It cannot therefore be too carefully borne in mind by 
Geologists that none of those numerical estimates, which relate 
to time, are applicable to the case of a liquid interior. 

With respect to the obliquity of the ecliptic, it seems probable 
that it may have originated when the moon broke away from the 
earth, however much the amount of it may have since changed ; 
for the rupture must have occurred at what was then the equator ; 
but the alteration in the principal axes of the earth owing to its 
removal must have caused the axis of rotation to shift its place 
within the mass, so that the plane of the moon's orbit would 
represent that of the original equator, while the plane of the new 
equator would have become oblique to it. 

Although, as just mentioned, the amounts of heat generated 
in the earth during certain intervals of time depend upon a 
particular assumed value for the viscosity, not so the whole 
amount since the rupture. Darwin says "According to the 
present hypothesis [of the generation of the moon] looking for- 
ward in time [from that epoch], the moon-earth system is from 
a dynamical point of view continually losing energy from the 

* Phil. Trans. Roy. Soc, vol. 14.5, p. 101. See also a lecture by Sir G. B. Airy 
"On the interior of the Earth." Nature, vol. 18, p. 41, 1878. 

t See "Precession of a viscous spheroid and remote history of the earth." 
Phil. Trans. Pt. n. p. 531, 1879. 



340 Rev. 0. Fisher, On the hypothesis of a [May 30, 

internal friction. One part of this energy turns into potential 
energy of the moon's position relatively to the earth, and the 
rest develops heat in the interior of the earth." It is evident 
therefore that, knowing the initial and present circumstances, 
it is possible to estimate the total amount of energy converted 
into heat without knowing the lapse of time in which it has 
occurred. Darwin finds the common period of rotation, when 
the moon separated from the earth, to have been 5 h. 36 m., 
taking the viscosity at that time as small, the earth being sup- 
posed to have been " a cooling body gradually freezing as it cools." 
The present rate of rotation relative to the moon (the lunar day) 
is 23 h. 56 m. The total heat generated in the earth in the 
course of this lengthening of the day if applied all at once would 
he says* be sufficient to heat the whole mass of the earth about 
3000° Fah. supposing it to have the specific heat of iron. In 
Table iv. of the former paper -f- he had given 1760° Fah. as the 
temperature corresponding to a period of rotation of 6 h. 45 m., 
so that it appears that the additional 1240° must be due to 
the loss on the difference between 6 h. 45 m. and 5 h. 36 m., 
or 1 h. 9 m., and he remarks that, "The whole heat generated 
from first to last gives a supply of heat at the present rate of 
loss for 3560 million years. This amount of heat is certainly 
prodigious, and " he adds, " I found it hard to believe that it 
should not largely affect the underground temperature"^; but a 
further calculation led him to believe that it need not do so, 
for he found that 0"32 of the whole heat would be generated 
within the central eighth of the volume of the earth, and only 
one-tenth within 500 miles of the surface. The heat generated 
at the centre is 3 T 7 ¥ times the average, that at the pole 1/2^ 
of the average, and at the equator 1/12| of the average; and it 
turned out that the heat, being so centrically produced, would, on 
account of the slowness of conduction, not have had time to reach 
the surface in the 57 million years postulated. This conclusion 
depending on conduction would of course be true only in the case 
of a solid earth, the interior of which had the particular viscosity 
which has been assumed, on which the 57 million years depend. 

In connection with this point a serious difficulty seems to 
arise. Lord Kelvin, in his well-known paper "On the secular cooling 
of the Earth §," held that, when according to his view it solidified 
in a comparatively short period of time, the interior was at the 
temperature of solidification suited to the pressure at every depth, 
and, because the cooling would not even yet have penetrated to 

* "Problems connected with the tides of a viscous spheroid," p. 592. 

t "On the precession of a viscous spheroid," p. 494. 

X p. 561. 

§ Trans. Boy. Soc. Edin. vol. xxm. pt. i., p. 157, and Nat. Phil., App. D. 



1892.] liquid condition of the Earth's interior. 311 

any great depth, it ought to be so still if it is solid. How 
then, it may be asked, could this enormous amount of heat be 
perpetually being communicated to the central parts, and they 
still remain solid ? It seems that they must have become heated 
far above the temperature of fusion appropriate to the pressure, 
and must now be liquid ; as nearly all geologists believe. 

I think I have proved in the Physics of the Earth's Crust* 
that, if the crust is as thin as geologists suppose, and if the age 
of the world is anything approaching to what geological pheno- 
mena appear to indicate, then there must exist convection currents 
in the interior, which prevent the crust from growing thicker 
by melting off the bottom of it nearly as fast as it solidifies. 
But I made no suggestion to account for such currents being 
maintained. Here however we appear to find the explanation. 
This centrically generated heat would be amply sufficient to 
support fusion, and to keep the currents in action. Indeed the 
difficulty is rather to see what would become of it all. Darwin's 
result, regarding the localization of the heat generated, does not 
depend upon the viscosity, for the coefficient (v) which is intro- 
duced into the calculation does not appear in the final result - }-; 
but it applies only to the heat generated within the earth by 
the action of the tidal couple upon the substance of the interior. 
The distribution of heat within the earth caused by the tidal 
couple will still follow the same law if only a portion of it is 
generated within the earth, and the rest within the water of the 
ocean. Suppose for instance that the earth was either perfectly 
rigid or perfectly fluid. In either such case no heat would be 
generated within the earth. But without doubt the friction of 
the oceanic tidal flow would, in a sufficiently long time, reduce the 
speed of the rotation |. The heat in that case would be generated 
only in the water, and be radiated into space. But besides friction 
there seems reason to believe that some amount of heat may be 
generated in the ocean owing to the fact that the speed of the 
forced tide wave differs from that of the free wave with which a 
disturbance, would travel round the earth under the influence of 
gravity alone. The question is an interesting one, and the following 
attempt is made to solve it. 

We have cf> the west longitude of P the place of observation, 
cot the moon's angular distance west of the prime meridian. Then 
the moon is cf) — cot east of P. 

* 2nd Ed. pp. 77 and 349. 

t " Problems connected with tides of a viscous spheroid," p. 558, equation (28), viz. 

where H is the average loss of heat throughout the earth. 

J Sir W. Thomson on "Geological Time." Trans. Geol. Soc. of Glasgow, 
vol. in. pt. i., 1868, p. 6. 



342 



Rev. 0. Fisher, On the hypothesis of a [May 30, 



Suppose, as is usual in the canal theory, that AP is developed 
into a straight line, and that the earth is at rest, and the moon 




moving westward above AP. Then, if AP = x, the attraction 
of the moon on the water at P will be in the direction to diminish 
x, and will be negative. The moon's differential horizontal attrac- 
tion at P will therefore be 

daM . 0/ , . 
- -22J3- sm 2 (4> - at \ 

which for shortness write — fi sin 2-v/r. 

Let, as before, the depth of the canal be h, and its width 
unity, i.e. one foot, and let y be the height of the tide above the 
undisturbed water. 

Then we have for the accelerations on a unit particle of water 
atP 

X = — fj, sin 2-yjr, 
Z = -g. 
And x = a^r, .'. dx = ad\}r. 

Now the work on a unit particle 



= p I {Xdx + Zdz) 

= p I (— /xa sin 2-^rd^r—gdz) 

= p (^cos2f-gz"j + C. 



1892.] liquid condition of the Earth's interior. 343 

At an angular distance of 45° from the moon the water is 
at rest, and its depth is the mean depth of the canal, viz. h. This 
makes z the depth at P ; 

.-. = -pgh + C. 

Hence the work on a unit mass of water in the column at 
Pis 



p ^cos2f- g(z-h) 

= P ( y cos 2 ^ ~ 9y) 



= p (^ cos 2f + g —M cos 2f[ 

pa a 2 co 2 _ . 

= P ^-tt ^—9 T COS Z-vZr. 

H 2 a'co'-gh r 

Therefore the work on the whole column of unit width is 
pa a 2 co 2 a ,, s 

pa a 2 co 2 a , fi P a h o , N 

= P ^ -^—5 ; COS 2^1r [h ■— 7j- V^ f cos 2 T 

^ 2 aW-gh r \ 2 aW-gh T j 

(pa , a 2 to 2 o , />a aw \ 2 A /1 .} 

= P1^' l -r^ r COs2i|r- [ r ~ -^— z -(l + COs4-f)k 

r [ 2 aw—gh T \2 a'to—ghj 2 X r 7 J 

To obtain the work done on a length J.P of the canal we must 
multiply this by acZ-v/r and integrate, whence, putting for -yfr its 
value ^ — &)£, and taking the integral from t = to £ = £, we get 



work on AP — p 



t-r- h -s— » , {sin 2(6- cot) — sin 2(6] 

4 aco—gh 1 - ^ r> 



(pa aco \ /ia 
+ lT«V^AJ 2 w * 



1 Aua aaj \ 2 ha , . . , , jX . . , ' 






The work on the whole canal will be given by putting (6 = 27: , 
and will be, 

work on the whole canal = p \——t- h—»-T, j sin 2cot 

r { 4 aV - gh 

(pa aa> \ 2 ha 

+ {Y aW-gk) ~2 at 

1 (pa aco \ 2 ha . . ,) 

4 V 2 aV - gh 2 ) 



844 Rev. 0. Fisher, On the hypothesis of a [May 18, 

This work is done by the moon upon the whole mass of water, 
while she traverses the interval AQ. 

Hence the work clone while she makes a complete revolution 
will be given by putting a>t = 2ir, and it will be 

ffia aco \ 2 ha 

p \jM=gh) T 2lr ' 

This work will accumulate once every lunar day. 

To obtain the corresponding rise of temperature, we know that 
a weight m raised through s feet is equivalent to heat sufficient to 
warm m pounds of water through s/772 degrees Fah. ; so that to 
find the rise of temperature produced by the work W upon a mass 
m of water we have 

W = mgs, 

and the equivalent rise of temperature in the water will be 

Wmg 

In the present instance 

m — plirha, 

and therefore the rise of temperature in the water in a lunar day 
will be 

/fia aw \ 2 1 

V "2" aV - gh) 2#772 " 



We know that 



Ma? _ _1 

ED 3 ~ 18-2 x 10 6 



Hence ^ = ti8^T0^- 

And « = 0-000072924 radian, 

a = 3959 miles, 

h = 4 miles, 

g — 32 feet per second. 
Reducing to feet, the rise of temperature in the water of the 
equatorial canal in degrees Fah. comes out about 

0-000006° Fah. 
in one year, or 6° in a million years. 

We see then that under the present circumstances a very 
small portion of the heat generated about the earth in this manner 
would be taken up by the ocean, and radiated into space, irre- 
spective of the friction of the water. But Darwin informs us 

* Thomson and Tait, 2nd Ed. p. 383. 



1892.] liquid condition of the Earth's interior. 345 

that, looking backwards, the moon's orbital velocity increases very 
rapidly. Now co is the earth's angular velocity minus the moou's 
orbital velocity. If then retrospectively the moon's orbital velocity 
increases more rapidly than the earth's angular velocity, co will 
diminish. 

11 we put u = ' 



\2 aW-gh, 

, du _ ffjLCiV 2ar co (a 2 co 2 + gh) 

then dco - ~ [TJ (aW-ghf • 

Hence, so long as a 2 co 2 is greater than gh, u will increase as co 
diminishes. Moreover the moon's distance diminishes. Hence, 
(fj.a/2) 2 varying inversely as the sixth power of the distance will 
increase very rapidly; so that on both these accounts the heat 
generated in the water per lunar day will rapidly increase. It 
must not however be forgotten that the length of the lunar day 
increases, so that fewer of them go to a year. 

The above expression would become infinite if 

co = s/ghja ; 

that is if co = 0-000039, 

whereas at the present co = 0*00007.3. 

But such a result cannot be relied upon, because the same value 
of co would make the expression for the height of the tide infinite, 
whereas in the formation of the differential equation from which it 
is found it has been assumed to be small. But it is evident that 
the generation of heat in the water must increase as that value of 
co is approached, and that something of the nature of a catastrophe 
will have happened at that juncture, because, going back in time, 
when that epoch has been passed the expression for the height of 
the tide is found to have changed signs, and consequently high 
and low water will have interchanged places then. 

If co is less than vgh/a, then du/dco becomes positive, and the 
heat generated in the water rapidly diminishes as co diminishes. 

We know that "Jgh is the velocity of the free wave, with which 
a disturbance in the water would be propagated under the influ- 
ence of gravity alone. 

The friction of the tides against the coast-lines will of course 
have had some effect in retarding the rotation, but how much we 
cannot estimate*. 

"We have seen that the fact that the speed of the forced tide 
wave in the ocean differs from that of the free wave is a cause of 

* Airy's " Tides and Waves," § 544. Encycl. Met. quoted by Sir W. Thomson, 
*'Geol. Time." Trans. Geol. Soc, Glasgow, 1868. 



346 Rev. 0. Fisher, On the hypothesis of a [May 30, 

generation of heat (though small) in the water. A like cause 
must be in operation within the earth, because the forced bodily 
tide has a different period from the free gravitational oscillation. 
The distribution of the additional heat from this cause would pro- 
bably, if calculated, turn out to be different from that arising from 
the internal friction produced by the tidal couple, which is the 
source of internal heat contemplated in Prof. Darwin's work. 

It seems then that, unless the ocean tides have been in opera- 
tion for a length of time exceeding any estimate hitherto suggested, 
it does not appear probable that any considerable portion of the 
heat, which according to Darwin's hypothesis of the moon being 
shed from the earth has been from first to last generated about 
the earth, can be got rid of by that means. It follows that a 
largely preponderating amount of it must have accumulated within 
the earth. This as already remarked must have kept the deeper 
parts constantly above the temperature of solidification for the 
pressure, and is an argument in favour of present liquidity. 

But if such is their condition we cannot appeal to the slowness 
of conduction in a solid earth to account for this great amount of 
heat not making itself evident at the surface ; which it must have 
done unless it has been prevented from accumulating faster than 
it has been generated. There seem to be only three important 
means of effecting this, viz. (1) conduction through the solidified 
crust, (2) transference of heat to the surface by volcanic action, 
(3) the conversion of heat into work against gravity, and against 
the molecular forces, expended in modifying during geological 
ages the condition of the crust. 

The first and most obvious mode of escape of heat from the 
interior, which we now regard as liquid, is by conduction through 
the solidified crust. I have explained in my Physics of the Earth's 
Crust*, how it is the latent heat of the layer by which the crust 
would have been thickened more than, owing to the action of the 
hot liquid, it is actually thickened, which escapes by conduction 
through the crust, and that this heat is abstracted from the interior 
mass and lowers its temperature. I have also shown that the 
mean fall of temperature of the interior from this cause, consider- 
ing the store of heat to have been initial f and the time elapsed 

* p. 73. 

t It appears however that this assumption is not necessary, for the whole mass 
remelted (using the symbols in Physics of the Earth's Crust) is y x Air (r - k) 2 , and 
it yields up X times that amount of heat. 

This divided by the volume of the interior will give the mean fall of temperature 

!*(?•- 7c) 3 J r-k 

or putting y for -- = 3X7 - nearly, 

as on the hypothesis of the heat being initial. 



1892.] liquid condition of the Earth's interior. 847 

100 million years, may be put at about 209° Fab. This calculation 
involves the assumption that the ratio of the rate of thickening to 
the rate of retardation (or remelting) is constant, or, what is equi- 
valent to this, that the thickness of the crust varies as the square 
root of the time since it began to be formed. It was there shown* 
that the assumption of constancy of the above ratio of the rates of 
thickening and retardation of thickening is probable, because, if 
their ratio varied as any power of the time, it would lead to 
unnatural consequences. I now find that the assumption that the 
store of heat was initial is not necessary, because the same formula 
can be obtained without that assumption, so long as we adhere to 
the other assumption that the thickness of the crust varies as the 
square root of the time, or to its equivalent. Such a calculation 
is sufficient to show that the amount of heat, carried off by con- 
duction through the crust in an interval even so long as 100 million 
years, must have been quite inconsiderable compared to the whole 
amount generated in the interior. 

As the most extreme case possible of volcanic action we can 
estimate approximately the fall of temperature of the earth sup- 
posing the whole of the water of the ocean to have been originally 
in solution with the magma of the interior, and to have carried off 
a corresponding amount of heat. Professors Riicker and Roberts- 
Austen have determined the temperature of melting basalt to be 
about 920° C. f Now the total heat of steam at t degrees C. given 
off in condensing into water at 0° C. is given by the formula 

605-5 +0-305*+; 

whence it appears that unit of vapour at 920° C. will have parted 
with 886 units of heat in condensing to water at 0° C. Hence every 
unit mass of the ocean on the hypothesis now made represents 886 
units of heat removed from the interior of the globe ; for remem- 
bering that the main body of water in the great oceans is at very 
low temperatures, and that a large volume of water at the poles is 
frozen, it is not a violent supposition to assume 0° C. as the mean 
temperature. 

Now even supposing the ocean to cover the globe and to be four 
miles deep, its volume will be about 0'003 of the whole globe. The 
density of the globe is 5*5 that of water. Hence the mass of the 
ocean is 0003/55 times the mass of the globe. Then, taking as 
Darwin has done the specific heat of the globe to be that of iron, viz. 
1/9, we get the mean temperature of the interior reduced by this 
means by 4°"53 C. or 8° F., an inappreciable amount compared with 
the 3000° F. attributed to tidal action, by which the earth is 
estimated to have been heated. 

* Appendix to Physics of the Earth's Crust, p. 21. 

t " Nature," vol. xliv., p. 456. Also Phil. Mag., Oct. 1891. 

% Tait's Heat, § 166. 

VOL. VII. PT. VI. 27 



348 Mr Willis, On Qynodioecism in the Labiatae. [May 30, 

On the above extreme hypothesis that the ocean consists of 
condensed steam emitted from the interior, the solid ejectamenta 
of volcanic action would have had a very subsidiary effect in re- 
ducing the internal temperature. 

There remains the consideration of the heat converted into the 
work which has been expended in producing elevations of the 
surface, in shearing and contorting the materials of the crust, and 
in inducing molecular changes. The amount of this work has no 
doubt been from first to last enormous ; but it is easy to see that a 
very inconsiderable fall of temperature throughout the interior 
would represent a very great deal of such work effected. For 
instance, the work of raising through half its height a layer of 
granite ten miles thick, weighing 178 pounds per cubic foot, would 
represent the heat equivalent to a fall of temperature of only 
one degree F. throughout the globe. 

We have not then so far arrived at an answer to the enquiry — 
What has become of all the heat generated by tidal friction ? 
There appear to be only two replies to this question. One is, that 
the solidification of the crust took place a very long while subse- 
quent to the genesis of the moon, so that the still liquid surface 
was able for ages to radiate directly into space the heat carried 
up to it from below by convection during the time, when, owing to 
the proximity of the moon, the generation of internal heat went on 
most rapidly. The other answer can only be, that the moon was 
not originally thrown off from the earth, but was left behind accord- 
ing to the nebular hypothesis. In that case the whole amount of 
tidal action would not have been so great, though nevertheless 
sufficient heat may have been centrically generated by it to main- 
tain those internal currents, which the theory of a thin crust and 
liquid interior appear to necessitate. 



(2) On Gynodicecism in the Labiatae. (First paper.) By 
J. 0. Willis, B. A., " Frank Smart " Student in Botany, Caius 
College. 

In July, 1890, my attention was called, by Mr F. Darwin, to 
the occurrence, on hermaphrodite plants of Origanum vulgare in 
his garden, of occasional flowers having one, two, three, or even 
all, of the stamens aborted. I found such flowers, on examination 
of many plants, to be of fairly common occurrence. The corolla is 
usually smaller than in the normal hermaphrodite flower, and 
may even, especially in the case of the female flowers, be as small 
as the corolla of a normal female flower (i.e. a flower on a female 
plant). The aborted stamens are represented by small dark- 
coloured bodies in the throat of the corolla, usually sessile, but in 
some cases shortly stalked. 



1892.] Mr Willis, On Gynodioecism in the Labiatae. 



349 



Corresponding variations on the normally female plants were 
of much rarer occurrence. Occasionally, however, I found a female 
plant bearing a large hermaphrodite flower among the females, 
and flowers with one or two stamens also occurred. 

That these variations are not simply due to cultivation appears 
from the fact that they are as common upon the wild form, which 
I have examined at Abington (Cambs.) and Llangollen. Three 
batches of plants gathered at the same time in August, 1890, gave 
the following results : 



Batch 


Plants 


Flowers 


I. 


II. 


III. 

14 
11 

6 


IV. 


Total 


% 


A. 
B. 
C. 


12 

7 
9 


1146 

745 
588 


8 
8 
2 


8 
21 
17 


37 
14 

7 


67 
54 
32 


5-85 
7-23 
5-44 


Total 


28 


2479 


18 


46 


31 


58 


153 


6-17 



A. Plants from the "Labiatae" bed, Bot. Gardens, Camb. 

B "Medicinal" 

C Abington (Cambs.) 

The numbers in columns I., II., III., IV., represent the numbers of flowers with 
1, 2, 3, 4 aborted stamens, respectively. 

It will be noticed that the number of flowers fully female is 
greater than the number of any of the intermediate forms, and 
this I found to be always the case, if a considerable number of 
plants were examined. Some of the variations were very striking, 
e.g. on each of two plants in batch A there occurred a lateral 
twig which bore female flowers only. I have observed the same 
phenomenon on two or three other occasions. 

Similar variations occur upon the hermaphrodite plants of 
other Labiatae. Miiller*, Schulz-|-, and others have observed them 
in some, and I have myself noticed them in Thymus serpyllum, 
Nepeta Glechoma, and N. Gataria (all in the wild state), besides 
many garden plants of the order, e.g. Micromeria Juliana, Nepeta 
longijlora, Hyptis pectinata, Bystropogon punctatus, Mentha crispa, 
Satureia hortensis and S. montana. The last-mentioned is ex- 
tremely variable, at least in Cambridge, more than half the flowers 
usually departing from the normal type. 

During 1891 various observations were made upon these 
abnormalities, with a view to discovering the conditions govern- 
ing them, and also to throwing some light upon the origin of 

* '• Fertilisation of Flowers," Eng. Ed. p. 476 (Galamintha Glinopodium). 
\ "Die biologischen Eigenschaften von Thymus Chamaedrys Fr. u. T. angusti- 
folius Pers." Deutsche Botan. Monatsschr. in. 1885, p. 152. 

27—2 



350 Mr Willis, On Gynodicecism in the Labiatae. [May 30, 

gynodioecism. If Lud wig's* view be correct, that the primary 
cause is the protandry of the flower, rendering the stamens of the 
earlier flowers useless, we might expect to find these variations 
more frequent at the commencement of the flowering season. 
This was tested as follows: Ten cuttings were taken from the 
same parent stock, and grown under similar conditions : every 
flower was carefully examined. The abnormalities were most 
erratic in their occurrence, and I was unable to discover any con- 
ditions governing this point. No two of the plants gave results 
corresponding in any way, nor did the average follow any ap- 
parent rule. Some bore about the same percentage of abnormal 
flowers throughout the season, others bore them many at one 
time, few at another: e.g. No. 1 bore 126 females altogether; 73 of 
these appeared in three days, and of these 29 were upon one small 
lateral branch of the inflorescence. It may however be noted that 
the percentage of abnormalities was much lower than in 1890, 
being only 2 per cent. ; a result possibly due to the plants 
being cultivated, and having no competition with one another for 
space. 

One plant was protected from insects by a muslin net through- 
out the flowering season, and did not, though it bore hundreds of 
flowers, set a single seed capable of germination. It should be 
noted, that owing to the smallness of the meshes, the plant could 
not be shaken by the wind. 

Observations were also made (1891) upon Nepeta Glechoma 
(wild). Two lots of plants were examined, one (A) growing on a 
dry sunny bank, the other (B) in deep shade in a wood. The 
latter commenced to flower 18 days later than the former and 
were much taller and less hairy. 

The numbers of plants in flower and the numbers of open 
flowers were counted weekly during the flowering season, and it 
was found, as Ludwig has observed in the case of thyme, that the 
proportion of female to hermaphrodite plants in flower was greater 
at the beginning than at the end of the flowering season. For 
example in lot A, on the first day 6 female plants and one 
hermaphrodite flowered ; the percentage of females being 857. A 
week later it was 33'6 per cent., and near the end of the season 
was 236 per cent. In lot B, the percentages each week were 50 - , 
16', 35-8, 28-5, 23-4, 19*2, 28-3. 

It was noticed that the female plants generally bore more open 
flowers at one time than the hermaphrodites. In lot A the 
number of flowers on each female plant was (on an average for the 
whole season) 2*40, and on each hermaphrodite 216. In lot B 
the numbers were 3'15 and 2"16. The greater size of the her- 

* " Ueber die Bliithenformen von Plantago lanceolata L. und die Erscheinung 
der Gynodiocie." Zeitschr. f. d. Ges. Naturw. Lit. 1879. p. 441. 



1892.] Mr Willis, On Gynodicecism in the Labiatae. 351 

maphrodite flowers is thus to some extent compensated for by the 
greater number of the females. 

Abnormalities in the flowers of Nepeta, like those observed in 
Origanum, &c, are comparatively few and far between, but were 
yet fairly often encountered. 

During the course of these observations upon Nepeta an 
interesting point was noticed. The protandry of the flowers 
appears to vary according to the season : at the beginning of the 
flowering season the stigmas begin to separate very soon after 
the dehiscence of the anthers, while towards the end of the 
season these processes are separated by a considerable time. I am 
conducting further observations upon this point. If it should 
prove general, it would, taken together with the negative results 
of the above observations on Origanum, have a tendency to dis- 
prove Ludwig's view of the origin of gynodicecism. This point 
however I hope to discuss in a future paper, when I shall have 
concluded the further observations on Origanum, &c, which are 
now beins conducted. 



(3) On the Steady Motion and Stability of Dynamical Systems. 
By A. B. Basset, M.A., F.R.S., Trinity College. 

1. The object of the present paper is to develop a method for 
determining the steady motion and stability of dynamical systems, 
by means of the Principle of Energy, and the Theory of the 
Ignoration of Coordinates. The subject has already been discussed 
by Routh*, but is treated in the present paper in a slightly 
different manner. 

Let the coordinates of a dynamical system consist of a group 0, 
and a group of ignored coordinates ^ ; and let k be the constant 
generalized momentum corresponding to %. Then if the velocities 
<% be eliminated by means of the equations 

dT_ 

dx~ K ' 

it is well known that the kinetic energy of the system will be of 
the form 

T=% + ®, 

where X is a homogeneous quadratic function of the velocities 6, 
and $ is a similar function of the constant momenta k. 

Also if © be that portion of the generalized component of 
momentum corresponding to 6, which does not involve 6, and 

* Treatise on Stability of Motion. 



352 Mr Basset, On the Steady Motion [May 30, 

which is consequently a linear function of the ac's, the modified 
Lagrangian function is 

L = %+Z(®0)-®-V (1), 

where V is the potential energy, measured from a configuration of 
stable equilibrium*. 

The equations of motion of the system are accordingly 

ddX d® d% *(d®A\,d$ dV , 

dtTe + ~dt~^e~ \dd V i + dd + W "° {Z) - 

From this equation it appears, that a steady motion may 
usually be obtained by assigning constant values to the coordinates 
; whence the equations of steady motion are 

de + d0~ l) {3) ' 

where the number of equations of the type (3) is equal to the 
number of coordinates 6. 

2. Let there be to coordinates of the type 0, and n ignored 
coordinates of the type % ; then we have three cases to consider, 
according as to is equal to, less than, or greater than n. 

Case I. to = n. 

In this case, the number of equations of the type (3) is equal 
to the number of momenta k ; hence these equations are sufficient 
to determine these momenta. Accordingly the conditions of steady 
motion are, that it should be possible, without violating the con- 
nections of the system, to assign constant values to the 0's, such 
that the values of the to momenta k, furnished by the solution of 
(3), should be real. 

Case II. m<n. 

In this case, the number of equations of the type (3) is less 
than the number of momenta k. It will therefore be usually 
possible, when the values of the 0's are given, for the momenta to 
possess a series of arbitrary values, which lie between certain 
limits. 

Case III. m>n. 

In this case, the number of equations of the type (3) is greater 
than the number of momenta k; it will therefore be possible to 
eliminate the momenta from (3) in one or more ways. Hence in 
order that steady motion may be possible, it will be necessary that 

* Proc. Gamb. Phil. Soc, vol. vi. p. 117; Basset, Hydrodynamics, vol. i. p. 174; 
where, in equation (36), the sign of V ought to be changed. 



1892.] and Stability of Dynamical Systems. 353 

certain relations should exist between the 0's, or that some of 
these quantities should have certain definite values. 

3. As an example of Case III., let us consider the steady 
motion of an ellipsoid, which is rotating about its centre of inertia 
under the action of no forces. In this case, there is one ignored 
coordinate, viz. ■xjr ; and the momentum corresponding to yjr is the 
constant angular momentum k about OZ. The value of §t is 

* = C cos 2 8 + (A cos 2 cj> + B sin 2 </>) sin 2 6 ^' 

The equations of steady motion are 

d$ =0 - # =0 (0) - 

which are satisfied 

(i) by = 0; 

(ii) by 6 — J?r, and = 0; 

(iii) by 6 — ^tt, and </> = \ir. 

These three conditions respectively correspond to rotation about 
the least, greatest and mean axes. Hence steady motion is 
impossible, unless the axis of rotation is a principal axis ; which 
is a well-known result. 

4. As an example of Case II., we may consider the motion of 
a solid of revolution or top, spinning about its point. Here <p, as 
well as y\r, is an ignored coordinate ; the constant momentum 
corresponding to <j>, is the angular momentum Cco 3 of the top about 
its polar axis. If k v k 2 be the momenta corresponding to yfr, (f>; 
the value of 5? will be found to be 

^_ ( /c i- /g 2 cos61 ) 2 , *«' (a\ 

h ~ 2Asin 2 6 + 2G K h 

also V=Mga(l + eos0) (7); 

whence the equation of stead) 7 motion is 

^(ff+TO = (8), 

and will be found to lead to the usual result. 

5. We must consider the condition of stability. 

Let E be the energy in steady motion, and let the suffixes 
denote the values of the quantities under these circumstances. 
Then 



354 Mr Basset, On the Steady Motion [May 30, 

Let any disturbance be communicated to the system, and let 
E + BE be the energy of the disturbed motion ; then 

E+8E = X + $+V, 
whence 

8E=Z + [(® + V)-(® +V )] (9). 

Now X, being the kinetic energy of a possible motion of the 
system, is essentially positive, and in the beginning of the dis- 
turbed motion must be a small quantity ; hence if 

$+V>® +V , 

% must remain a small quantity, and the system cannot deviate 
much from its position in steady motion, and the motion will 
be stable ; but if 

®+v<$ +v , 

% may become a finite positive quantity, whilst the term in square 
brackets may become a finite negative quantity, such that their 
difference remains equal to the small quantity BE, and the motion 
may be unstable. Hence the motion will be stable, provided 
i? + V is a minimum. 

It should be noticed, that this criterion of stability not only 
includes disturbances which produce variations of the coordinates 
6, but also disturbances which produce variations of the momenta 
k, though of course the latter quantities always remain constant 
during the disturbed motion, and equal to their values immediately 
after disturbance. 

6. Routh has shown*, that when there is only one coordinate 
of the type 0, the steady motion will be unstable unless £ + V is a 
minimum in steady motion ; but although we have just shown, 
that when there are two or more coordinates of the type 6, the 
motion will be stable provided $ + V is a minimum, it does not 
follow that the motion will be unstable, when this condition is not 
satisfied. In fact it sometimes happens, that steady motion will 
be stable when £ + V is a maximum. To see this, let us re- 
turn to the case of the ellipsoid rotating about its centre. 

When the axes of rotation are the least, greatest and mean 
axes, the values of £ in the beginning of the disturbed motion 
are respectively equal to 

2 /c 



C - {(G - A)cos 2 cf> + (G - B)sin 2 cf>}6 2 ' 
A + (B-A)<f> 2 + (C-A)e 2 ' 

2 K 



B-(B-A)f 2 + (G-B)e 2 ' 

* Stability of Motion, p. 85. 



1892.] and Stability of Dynamical Systems. 355 

where 0, <p, e,f are small angles. In the first case, 5? is a minimum 
in steady motion. In the second case, in which rotation takes 
place about the greatest axis, £ is a maximum ; but it can be 
shown by other methods, that the steady motion is stable. In the 
last case, in which the rotation takes place about the mean axis, 
£ is a maximum for some disturbances, and a minimum for others; 
and the motion is well known to be unstable. 

The conditions of stability, when there are two coordinates 
of the type 0, are given by Routh*, and are somewhat com- 
plicated. The general conditions of stability, when £■ 4- V is 
not a minimum in steady motion, do not appear to have been 
investigated. 

7. The advantages of the Theory of the Ignoration of Co- 
ordinates is most strikingly illustrated in Hydrodynamical pro- 
blems; for in this subject, the most convenient form of the 
kinetic energy is frequently one, which is not entirely composed 
of velocities, which are the time variations of coordinates. More- 
over, the generalized velocities corresponding to such quantities 
as components of molecular rotation, vorticity and the like, are 
frequently unknown, or would be troublesome to introduce. We 
shall presently call attention to two Hydrodynamical problems, 
in which the power of this method is strongly brought out. 

It must however be noticed, that it is not necessary that 
the quantities k should be momenta in the ordinary sense of 
the word; for if r t , t 2 ... be any other quantities, which are con- 
nected with k v k 2 ... by a series of relations of the form 

K i=fA T i> T 2> ••• )■ 

equations (2) and (3) would still apply ; provided the functions 
f do not contain any of the coordinates 0. This remark is of 
some importance, as a convenient transformation often shortens 
the work. 

8. In the case of a cylinder moving parallel to a fixed wall, 
when there is circulation f, the kinetic energy is 

T = 1R (x 2 + f) + K 2 paj4!7r, 

where k is the circulation. 

The coordinate x is an ignored coordinate, and the constant 
momentum h, corresponding to x, is the momentum of the system 
parallel to the wall, which is equal to 

h = Rx + icpc ; 

* Stability of Motion, p. 88. f Hydrodynamics, vol. i. § 213. 



356 Mr Basset, On the Steady Motion [May 30, 

whence £= ( ^^ + ^, 

and V=(M-M')gy. 

The equation of steady motion is 

4<a+F)-o. 

which leads at once to 

pu 2 - /cpu coth a + /c 2 /)/4ttc + (if - M ') # = 0, 

where P~ ~ h dRjdy, u = ob, 

which is equation (10), § 213. 

The stability of the various cases which arise, can be found 
from the condition that 

Jp (ffi+F) 

should be positive, but the calculation would be somewhat trouble- 
some. If however the radius of the cylinder were small in com- 
parison with its distance from the wall, approximate results might 
be obtained in a fairly simple form. 

9. In Hydrodynamical problems, which involve molecular 
rotation, certain quantities occur, which although not properly 
speaking momenta, are quantities in the nature of momenta ; and 
to this class of quantities vorticity belongs. Let a surface 8 
be drawn in a liquid, cutting each vortex line once only ; let co 
be the resultant molecular rotation, e the angle between the 
direction of &> and the normal to dS drawn outwards. Then the 

integral 

r f 

o) cos edS 



//■ 



is constant throughout the motion, and this statement expresses 
the fact that the vorticity of the mass of liquid is constant. The 
dimensions of this quantity, when multiplied by p are \ML 2 T~ 1 \ 
and these are the dimensions of an angular momentum. We 
may therefore regard the constant quantity 

p \\ a cos edS, 

as a generalized component of momentum. 

When a liquid ellipsoid is rotating about a principal axis 
(say c), the value of this integral is 

irpab^= 7rpR 3 ^fc, 



1892.] and Stability of Dynamical Systems. 357 

where R is the radius of a sphere of equal volume ; whence 
£/c must be constant throughout the motion, and we may there- 
fore treat this quantity, as one which is in the nature of a 
generalized component of momentum. 

The angular momentum h about the axis is also constant ; 
and if we introduce two new constants r, t, such that 

J/ c = ( T ' - r)/2abc, h = \M <Y + r) ; 

it follows, that since the volume is constant, we may regard 
t', t as quantities in the nature of generalized components of 
momentum, and employ them in the place of £/c and h. Whence 
the expression for $ is 



{(a-by^(a + bf 
and 

accordingly the steady motion is determined by the equations 

in which c must be regarded as a function of a and b. See 
Hydrodynamics, vol. II. §§ 363—367. 



INDEX TO VOLUME VII. 



Absorption of Energy by the Secondary of a Transformer, 249. 

Acacia latronum, 65. 

Acacia sphaerocepkala, Germination of, 65. 

Adami, J. G., Elected Fellow, 35. 

On the action of the Papillary Muscles of the Heart, 78. 

On the disturbances of temperature which follow extirpation of the 

fore-brain, 156. 
Agelena labyrinthica, the Oviposition of, 97. 
Airy, Sir G. B., 132, 134. 

Alcock, Miss E., The Digestive Processes of Ammocoetes^ 252. 
Alcyonium digitatum, 305. 
Allbutt, T. C., Elected Fellow, 327. 
Amici's Prism Telescope, 85. 
Ammoccetes, Digestive Processes of, 252. 
Annual General Meeting, 1, 93, 249. 
Associates, 100. 
Astacus, 257. 

Astronomy, A Model to illustrate facts in, 125. 
Astronomy, recent advances in, 330, 

Baker, H. F., Elected Fellow, 21. 

On the concomitants of three ternary quadrics, 32. 

Basset, A. B., On stabibility of dynamical systems, 351. 
Bateson, Miss A., On Variations in Floral Symmetry, 96. 
Bateson, W., On the perceptions and modes of feeding of Fishes, 42. 

On Skulls of Egyptian Mummied Cats, 68. 

Supernumerary Appendages in Insects, 159, 219. 

Variations in the Colour of Cocoons, 251. 

Berry, A, Elected Fellow, 21. 

Bipalium Kewense, 142. 

Blood-clotting, Experiments on, 163. 

Brady, H. B,, Elected Honorary Member, 100. 



360 Index. 

Brill, J., On Systems of Equipotential Curves, 126. 

Quaternions and Laplace's Equation, 120. 

On Points related to Families of Curves, 57. 

On Quaternion Functions, 151. 

Brindley, H. H., Elected Fellow, 100. 

On the size and number of sense-organs, 96. 

Brioschi, F., Elected Honorary Member, 99. 
Brown, E. W., Elected Fellow, 21. 

On the parallactic class of inequalities in the moon's motion, 220. 

Bryan, G. H., On a Problem in the Linear Conduction of Heat, 246. 

On the nodal lines of a revolving Cylinder or Bell, 101. 

Burnside, W., On the Theory of Functions, 126. 

Cats, Egyptian mummied, 68. 

Cayley, A., Non-Euclidian Geometry, 35. 

Cell- Granules, reaction of with Methylene-Blue, 256. 

Cells, On Clark's, 250. 

Chree, C, On changes of dimensions of elastic solids, 319. 

On Compound Vibrating Systems, 94. 

On Elastic Solids, 31. 

On liquid electrodes in vacuum tubes, 222. 

On long rotating cylinders, 283. 

On thin rotating isotropic disks, 201. 

Clark, J. W., Presidential Address, i — 1. 2. 

Clotting, intravascular, 308. 

Cocoons, Variations in the Colour of, 251. 

Cole, B. S., On a Linkage, 222. 

Comet, perturbations of, 314. 

Concomitants, 32. 

Conductivity, Effect of Temperature on, 137. 

Contacts of Circles and Conies, 262. 

Cooke, A. H., On the common Dog-Whelk, 13. 

On Parasitic Mollusca, 215. 

Coordinates, ignored, 351. 

Copper-Zinc Couple, 52. 

Council, 1, 34, 93, 129, 249, 282. 

Crookes, W., 223, 224, 225, 229, 243. 

Crystallization, On Solution and, 84. 

Cucurbitaceae, Stem of the, 14, 65. 

Cycas revoluta, 45. 

Cyclostomatous Polyzoa, Embryos in the Ovicells of, 48. 

Dana, J. D., Elected Honorary Member, 100. 
Daphnia, 257, 258. 

Darwin, F., On Kectipetality and the Klinostat, 141. 
Darwin, G. H., On genesis of the Moon, 335. 



Index. 361 

Darwin, G. H., perturbations of a Comet, 314. 

On Tidal Prediction, 151. 

Davy, Sir H., On an Experiment of, 250. 

Dawson, H. G., Elected Fellow, 156. 

Diffraction, at Caustic Surfaces, 131. 

Dog-Whelk, Varieties and Geographical Distribution of, 13. 

Earth, Rigidity of the, 72. 

Earth, liquid interior, 335. 

Elastic Plate, The finite deformation of, 31. 

Elastic Solids, 31, 319. 

Electric discharge, 314. 

strength, minimum, 330. 

Electric Discharge through rarified gases, 131. 

Electrical System, vibrating, and its radiation, 165. 

Electrical Waves in dielectric media, 164. 

Electricity, Contact- and Thermo-, 269. 

Electrodes, Discharge through rarefied gases without, 131. 

Electrolytes, Viscosity and Conductivity of, 21. 

Fahrenheit's Thermometrical Scale, 95. 

Favnlaria, 46. 

Ferments, injection of, 16. 

Fibrin-ferment, On the action of, 6V. 

Fisher, 0., On hypothesis of Earth's liquid interior, 335. 

Fishes, Perceptions and modes of feeding of, 42. 

Flaws, Effect of, on the Strength of Materials, 262. 

Floral Symmetry, Variations in, 96. 

Fluid Motions, Discontinuous, 175. 

Flux, A. W,, Elected Fellow, 100. 

Frog, Development of the Oviduct in the, 148. 

Functions, Theory of, 126. 

Gamgee, A., On Fahrenheit's Thermometrical Scale, 95. 
Gardiner, W., On the germination of Acacia sphaerocephala, 65. 
Geometry, Non-Euclidian, 35. 
Gibbs, J. W., Elected Honorary Member, 100. 

On Thermo-dynamics, 279. 

Glaisher, J. W. L., On the Series with pentagonal exponents, 69. 
Glazebrook, R. T., awarded Hopkins Prize, 21. 

On Clark's Cells, 250. 

Gold-Tin Alloys, 250. 

Greenhill, A. G., Strains in Shafting, 299. 
Groom, T. T., On the Orientation of Sacculina, 160. 
Gynodioecism in the Labiatae, 348. 



362 Index. 

Hankin, E. H., Elected Fellow, 35. 

On Vibrio Metschnikovi cultures, 311. 

A new result of the injection of Ferments, 16. 

Hardy, W. B., Reaction of certain Cell-Granules with Methylene-Blue, 256. 
Harmer, S. F., Embryos in the ovicells of Cyclostoniatous Polyzoa, 48. 

Excretory processes in Marine Polyzoa, 219. 

On Rhynchodemus terrestris, 83. 

Heart, Papillary Muscles of the, 78. 

Heat, Linear Conduction of, 246. 

von Helmholtz, 27, 28, 71, 165, 270, 279. 

Hertz, H., Elected Honorary Member, 100, 169. 

Heycock, C. T., On change of zero of thermometers, 319. 

Hicks, W. M., awarded Hopkins Prize, 92. 

Hickson, S. J., On Alcyonium digitatum, 305. 

The Medusae of Millepora, 147. 

Hill, G. W., Elected Honorary Member, 100. 
Honorary Members, 99. 
Hopkins Prize awards, 21, 92. 

Insects, Supernumerary Appendages in, 159, 219. 
Tonic Velocities, 250. 
Isotropic Disks, 201. 
Ixora, fertilization in, 313. 

Jenyns, L., 83, 146. 
Jets, Liquid, 4, 111, 264. 

Kanthack, A. A., Elected Associate, 327. 

On Vibrio Metschnikovi cultures, 311. 

Kirchhofp, 11, 169, 191, 196, 269. 

Klaassen, Miss H. G., The Effect of Temperature on Conductivity, 137. 

Klinostat, Modification of the, 141. 

Kronecker, L., Elected Honorary Member, 99. 

Lachlan, R., theorems on Bicircular Quartics, 87. 

Lamb, H., Awarded Hopkins Prize, 92. 

Langley, J. N., The action of Nicotin on the Crayfish, 75. 

Larmor, J., On the Curvature of Prismatic Images, and on Amici's Prism 

Telescope, 85. 
- — - Effect of Flaws on the Strength of Materials, 262. 

The Influence of Electrification on Ripples, 69. 

The Laws of the Diffraction at Caustic Surfaces, 131. 

■ A mechanical representation of a vibrating electrical system, 165. 

On the most general type of electrical waves in dielectric media, 164. 

On the Motions of Rigidly connected Points, 36. 

On the Spherometer, 327. 

Laurie, A. P., Elected Fellow, 48. 



Index. 363 

Laurie, A. P., On Gold-Tin Alloys, 250. 

Vehicles used by the old Masters in Painting, 48. 

Lea, A. S., On an Astronomical Model, 125. 

On Rennin and Fibrin-ferment, 67. 

Leahy, A. H., On distribution of Velocities among moving molecules, 322. 

Lemniscates and other Inverses of Conic Sections, 222. 

Liveing, G. D., On Solution and Crystallization, 84. 

Lomatophloios macrolepidotus, 43. 

Love, A. E. H., The finite deformation of a thin elastic plate, 31. 

On Sir W. Thomson's estimate of the Rigidity of the Earth, 72. 

On discontinuous Fluid Motions in two dimensions, 175. 

Lymph, Action of, on intravascular clotting, 308. 

MacBride, E. W., The Development of the Oviduct in the Frog, 148. 
Macdonald, H. M., The Self-Induction of two parallel Conductors, 259. 
Magnetometer for testing iron and steel, 330. 
Maxwell, J. Clerk, 95, 164, 165, 209, 210, 211, 212, 213, 214, 259, 260, 273, 

275, 279. 
Medxime of Millepora, 147. 

Metschnikoff, E., Elected Honorary Member, 100. 
Meyer, V., Elected Honorary Member, 100. 
Mollusca, Parasitic, 215. 
Monckman, J., Action of Copper Zinc Couple on dilute solutions of Nitrates 

and Nitrites, 52. 
Moon, Genesis of, 335. 

Moon's motion, parallactic inequalities in the, 220. 
Murphy, J. K., Elected Fellow, 330. 
Muscles, Papillary, of the Heart, 78. 

Newall, H. F., On recent advances in Astronomy, 330. 
Newton's description of Orbits, 4. 
Nicotin, its action on Crayfish, 75. 

Officers, 1, 34, 93, 249. 

Orr, W. M c F., Contact Relations of Circles and Conies, 262. 

Painting, Vehicles used by the old Masters in, 48. 

Parker, J., Theory of Contact- and Thermo-Electricity, 269. 

Pascal's Hexagram, 221. 

Pearson, K., On impulsive stress in shafting, 4. 

Peripattis, 250. 

Pertz, Miss D. F. M., On Rectipetality and the Klinostat, 141. 

Phymosoma, new species of, 77. 

Platts, C, Elected Fellow, 48. 

Poincar£, H., Elected Honorary Member, 99. 

Polyzoa, Excretory Processes in Marine, 219. 

VOL. VII. PT. VI. 28 



364 Index. 

Potter, M. C, On the increase in thickness of stem of Cucurbitaceae, 14, 65. 
Prismatic Images, 85. 

Quartics, Theorems on Bieircular, 87. 

Quaternions, Application of to the Discussion of Laplace's Equation, 120. 

Rectipetality, 141. 

Reinke's glands, 65. 

Rennin, On the action of, 67. 

Revolving Cylinder, Beats in the vibrations of, 101. 

Rhynchodemus terrestris, 83, 143. 

Richmond, H. W., On Pascal's Hexagram, 221. 

Rigidity of the Earth, 72. 

Ripples, Influence of Electrification on, 69. 

Roberts, T., Elected Fellow, 35. 

Ruhemann, S., Elected Fellow, 100. 

Sacculina, Orientation of, 160. 

Schuster, A., Elected Honorary Member, 100. 

Searle, G. F. O, On a compound Magnetometer, 330. 

On an experiment of Sir H. Davy's, 250. 

Sedgwick, A., On a Peripatus from Natal, 250. 
Sense-Organs of certain Animals, 96. 
Seward, A. O, Elected Fellow, 48. 

On Lomatophloios macrolepidotus, 43. 

Sharp, D., Elected Fellow, 100, 327. 

Sharpe, H. J., On Liquid Jets and the Vena Contracta, 4, 111. 

On Liquid Jets under Gravity, 264. 

Shaw, W. N., On Electrolytes, 21. 

Shipley, A. E., On Bipalium Kewense (Moseley), 143. 

On a new species of Phymosoma, 77. 

On Phylloxera vastatrix, 252. 

Shore, L. E., On intravascular Clotting, 308. 

Skinner, S., Elected Fellow, 259. 

Smith, W. Robertson, Elected Fellow, 330. 

Society, Foundation and early years of the, i — 1. 

Solution and Crystallization, 84. 

Spherometer, 327. 

Spiders, Spinning Apparatus of, 16. 

near Cambridge, 98. 

Stability of Dynamical Systems, 351. 

Stokes, Sir G. G., 11, 86, 137. 

Strains in long rotating cylinders, 283. 

Strength of Materials, Effect of Flaws on the, 262. 

Stress, impulsive in shafting, 4. 



Index. 365 

Taylor, C, On Newton's description of Orbits, 4. 

Temperature, Disturbances following extirpation of the fore-brain, 156. 

Effect of, on the Conductivity of Solutions of Sulphuric Acid, 137. 

Thermometers, change of zero in, 319. 
Thomson, J. J., 138, 222, 227, 231, 246, 273. 

Absorption of Energy by the Secondary of a Transformer, 249. 

On Electric discharge, 314. 

On minimum Electric strength of a gas, 330. 

On Electric Discharge through rarefied Gases without Electrodes, 131. 

Thomson, Sir W. (Lord Kelvin), 72, 165, 202, 277. 

Tidal Prediction, 151. 

Transformer, Absorption of Energy by the Secondary of a, 249. 

Treub, M., Elected Honorary Member, 100. 

Velocities, of moving molecules, 322. 

Vena Contractu, 4. 

Vibrating electrical system, 165. 

Vibrating Systems, Compound, 94. 

Vibrio Metschnikovi, 158. 

Cultures, 311. 

Warburton, O, Elected Fellow, 21. 

On the Oviposition of Agelena labyrinthica, 97. 

On Spiders near Cambridge, 98. 

On the Spinning Apparatus of Geometric Spiders, 16. 

Waves, Electrical in dielectric media, 164. 

Whetham, W. C. D., Method of measuring Ionic Velocities, 250. 

Willis, J. C, Elected Fellow, 327. 

On fertilization in Ixora, 313. 

■ On gynodicecism in the Zabiatae, 348. 

Young, W. H., Elected Fellow, 48. 



Cambridge: printed by c. j. clay, m.a., and sons, at the university press. 



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THE FOUNDATION AND EARLY YEARS 
OF THE SOCIETY: 

AN ADDRESS DELIVERED BY 
JOHN WILLIS CLARK, M.A. Trin. Coll. President, 

ON RESIGNING OFFICE, 

27 October, 1890. 

When the President of a Society lays down his office, it is 
usual that he should take a more or less extended view of the 
past history of the body with which he has been connected, 
thank his officers, and anticipate a brilliant future from the efforts 
of his successor. I have no wish to depart from these excellent 
precedents. I am using no empty phrase when I say that I felt 
it a distinguished, and not wholly deserved, honour to be chosen 
to succeed my dear friend Mr Coutts Trotter — one who by his 
intellectual attainments, by the breadth of his sympathies, and by 
his unwearied efforts to develop the scientific side of University 
training, was so distinctly marked out as the proper person to 
preside over a Philosophical Society. I, on the contrary, though 
the office which I have held for so many years in connection with 
the New Museums may have enabled me to be a promoter of 



ii Address of Mr J. W. Clark, President, 

science in others, have few claims to be called a man of science 
myself; and had I not been ably supported by the officers of the 
Society, the high reputation which we have so long maintained 
might have been somewhat tarnished by the appearance of a name 
so humble as mine in the list of Presidents. The Secretaries, 
however, have taken good care to provide our meetings with 
papers excellent in matter and varied in character ; and the 
Treasurer has been most energetic and successful in improving 
our financial position. 

The Society has now been in existence for rather more than 
seventy years ; and I may say, without fear of contradiction, that 
though our position is different from what it was in days which 
many persons can remember, we still hold our own in public 
estimation — still exercise a powerful influence in the University. 
The time has not yet come for the history of the Society to be 
written ; but it has occurred to me that it might be interesting to 
place on record a few notes respecting its origin, and the early 
years of its corporate life. 

In the Easter Vacation of 1819 Professor Sedgwick — who had 
been elected to the Woodwardian Chair in the May of the previous 
year — was taking a tour in the Isle of Wight, and collecting 
materials for his first course of lectures, which he delivered in the 
ensuing Easter Term. He was accompanied by Mr Henslow of 
St John's College, and as the two friends walked and talked they 
deplored the want of some place in Cambridge to which those in- 
terested in science might resort, with the certainty of meeting per- 
sons of similar or kindred tastes, and where they might learn what 
was being done abroad. Their " first idea," we are told, " was to 
establish a Corresponding Society, for the purpose of introducing 
subjects of natural history to the Cambridge students " ; and on 
their return to the University they wrote " to their respective 
friends for their encouragement and support 1 ." Easter had fallen 
late that year (11 April), and therefore the Easter Term would be 
short. Sedgwick, moreover, was fully occupied with his lectures. 
The idea too, was novel, and in those days novelty, especially 
when it took the form of a combination for the prosecution of 
something foreign to the normal course of study in the place, was 
sure to encounter disapprobation, if not active opposition. Delay, 
therefore, was unavoidable ; and it need excite no surprise that 
the Michaelmas Term was far advanced before the following notice 
was circulated, at the suggestion, it is said, of Dr E. D. Clarke, 
Professor of Mineralogy, who entered into the scheme with 
characteristic enthusiasm, and was always spoken of by Sedgwick 
as the founder of the Society. 

1 Memoirs of the Bev. John Stevens Henslow, by the Eev. L. Jenyns. 8vo. Camb. 
1862, p. 17. 



on resigning office, 27 October, 1890. iii 



Cambridge, Oct. 30, 1819. 

The resident Members of the University, who have taken their 
first degree, are hereby invited to assemble at the Lecture-Room, under 
the Public Library, at twelve o'clock, on Tuesday, Nov. 2, for the 
purpose of instituting a Society, as a point of concourse, for scientific 
com munications. 

Hon. Geo. Neville 1 . Rev. A. Carrighan. [Joh.] 
Bishop of Bristol 2 . — T. Jephson. [Joh.] 

Dean of Carlisle 3 . *Rev. J. Holmes. [Pet.] 

Dr Kaye 4 . * — W. Mandell. [Qu.] 

Dr Davy 5 . * — J. Hustler 15 . 

Dr Webb 6 . * — J. Brown. [Trim] 
Dr E. Clarke 7 . — C. Macfarlan. [Trim] 

Dr Haviland 8 . *W. Hustler 16 . 

Dr Ingle 9 . *Rev. J. Lamb 17 . 
*Prof. Monk 10 . — T. Hughes 18 . 

Prof. Cumming 11 . * — J. Evans. [Cla.] 

Prof. Sedgwick. * — G. Peacock 19 . 
Prof. Lee 12 . — F. Fallows 20 . 

R. Woodhouse 13 . — J. Whittaker 21 . 

Rev. T. Kerrich 14 . *— R. Crawley. [Magd.] 

1 Master of Magdalene 1813—53; Dean of Windsor 1845—53. 

2 Will. Lort Mansel, D.D., Master of Trinity 1788—1820; Bp. of Bristol 
1808—20. 

3 Isaac Milner, D.D., President of Queens' 1788-1820; Dean of Carlisle 1793 
—1820. 

4 John Kaye, D.D., Master of Christ's 1814—30; Bp. of Bristol 1820—27; of 
Lincoln 1827—53. 

5 Martin Davy, M.D., Master of Gonville and Caius 1803—39. 

6 Will. Webb, D.D., Master of Clare 1815—56. 

7 Edw. Dan. Clarke, LL.D., Professor of Mineralogy 1808—22. 

8 Joh. Haviland, M.D., Eegius Professor of Medicine 1817—51. 

9 Tho. Ingle, M.D. Pet. 

iy Ja. Hen. Monk, B.D. Trin., Begins Professor of Greek 1808—23; Bp. of 
Gloucester 1830—56. 

11 Ja. Cumming, M.A. Trin., Professor of Chemistry 1815 — 61. 

12 Sam. Lee, M.A. Queens', Professor of Arabic 1819—31. 

13 Bob. Woodhouse, M.A. Cai., Lucasian Professor 1820—22; Plumian Professor 
1822—28. 

14 Tho. Kerrich, M.A. Magd., Principal Librarian 1797—1828. 

15 Ja. Devereux Hustler, B.D. Trin. 

16 Will. Hustler, M.A. Jes., Kegistrary 1816—32. 

17 Master of Corpus Christi 1822—50. 

18 Tho. Smart Hughes, B.D. Emm., Christian Advocate, 1822. 

19 Geo. Peacock, M.A. Trin., Lowndean Professor 1836—58, Dean of Ely 
1839—58. 

20 Fearon Fallows, M.A. Joh., Director of the Observatory at the Cape of Good 
Hope 1820—31. 

2i Job. Will. Whittaker, M.A. Joh, 

a2 



iv Address of Mr J. W. Clark, President, 

Rev. H. Robinson 22 . J. Henslow 24 . 

*W. Whewell 23 . 

It is interesting to remark that the thirty-three persons who 
signed the above notice differed widely in their pursuits and 
opinions, and were drawn from eleven Colleges. Among them 
are six Heads, six Professors, (of Mineralogy, Geology, Chemistry, 
Medicine, Greek and Arabic), and eleven tutors, or assistant- 
tutors 25 . It is clear, therefore, that from the first there was nothing 
sectarian about the Society ; it represented no clique ; its sup- 
porters were not distinguished by any singularity of dress, de- 
meanour, or speech ; they merely recognised the need of extending 
the studies of the University in a scientific direction. 

No detailed report of the proceedings at this preliminary 
meeting was drawn up, but on the next day a brief memorandum 
was circulated in the University. It ran as follows : 

Cambridge, Nov. 3, 1819. 

At a Meeting of the Members of this University, which took place 
on Tuesday, November 2, in the Lecture-Room under the Public 
Library, in consequence of a requisition to that effect, signed by a 
number of distinguished Individuals of the different Colleges, the 
following Resolutions were carried unanimously : 

1. — That Dr Haviland be called to the chair. 

Proposed by Dr Clarke, and seconded by Mr Kerrich. 

2. — That a Society be instituted as a point of concourse for scientific 
communication. 

Proposed by Prof. Sedgwick, and seconded by Mr Robinson. 

3. — That a Committee be appointed, consisting of the following gentle- 
men, who shall report to all Members of the University desirous of 
belonging to the said Society, such regulations as shall appear to 
them to be proper for the proposed institution : 

Dr Kaye. 
Dr Clarke. 
Dr Haviland. 



Prof. Sedgwick. 
Mr Bridge. 
Mr Jephson. 
Mr Fallows. 



Prof. Farish. 
Prof, dimming. 

Proposed by Prof. Monk, and seconded by Mr Hughes. 

22 Hastings Eobinson, M.A. Joh. 

23 Professor of Mineralogy 1828—32; of Moral Philosophy 1838—55; Master of 
Trinity 1841—66. 

24 Joh. Stevens Henslow, M.A. Joh., Professor of Mineralogy 1822—28; of 
Botany 1825 — 61. Mr Henslow did not formally resign the Professorship of 
Mineralogy on obtaining that of Botany until the mode of election had been settled 
by Sir J. Bichardson's award. Memoirs of Henslow, ut supra, p. 29. 

25 To the names of these an asterisk has been prefixed in the above list, 



on resigning office, 27 October, 1890. 

banks of the Meeting be given to Dr H 
ct in the chair. 
Proposed by Prof. Farish, and seconded by Mr Hughes. 



4. — That the thanks of the Meeting be given to Dr Haviland, for his 
able conduct in the chair. 



N.B. It is requested that all those gentlemen who are desirous of 
adding their names to the Society previously to the next Meeting, will 
signify their intention to the Members of the Committee. 

The Committee to whom this important duty was entrusted 
must have set about their work without delay, for in less than a 
week the following " Regulations " had been drawn up. The paper 
containing them is endorsed: "Report of the Committee appointed 
to form the regulations of a Society to be instituted in this Uni- 
versit}^ for Philosophical Communication ; to be read at the first 
meeting of the Society, on Monday, November 15, at one o'clock, 
in the Lecture Room under the Public Library." 

Cambridge, November 8, 1819. 

At a Meeting of the Committee appointed to form the regulations 
of a Society, to be instituted in this University, for Philosophical 
Communication, it was resolved : 

1. That the Society bear the name of The Cambridge Philosophical 

Society. 

2. That this Society be instituted for the purpose of promoting 

Scientific Enquiries, and of facilitating the communication of facts 
connected with the advancement of Philosophy. 

3. That this Society consist of a Patron, a President, a Vice-President, 
a Treasurer, two Secretaries, Ordinary and Honorary Members. 

4. That a Council be appointed, consisting of the above-mentioned 

Officers, and five Ordinary Members ; three of whom constitute a 
Quorum : and that no person under the standing of M. A. be of the 
Council. 

5. That the Officers of the Society, with the exception of the Patron, 

be annually elected by Ballot. 

6. That Ordinary Members be chosen from the Graduates of this 

University by ballot ; their Election being determined by a majo- 
rity of two thirds of the Electors present. 

7. That any person desirous of becoming a Member, be proposed by 
three Ordinary Members ; and his name hung up in the Society's 
room, until the third meeting after the proposition has been made. 

8. That Honorary Members be proposed by six Ordinary Members, 

and balloted for accordingly. 



vi Address of Mr J. W. Clark, President, 

9. That the Meetings of the Society be held on a Monday, once in 

every fortnight during full term. The President to take the chair 
at seven o'clock, p.m. and to quit it at nine. 

10. That the business of each meeting be conducted in the following 

order : 

1. The minutes of the preceding meeting read and approved. 

2. Notices of new motions presented. 

3. Members proposed. 

4. Members balloted for. 

5. Motions on the minutes brought forward and determined. 

6. Miscellaneous business. 

7. Communications read and presents acknowledged. 

11. That all communications be sent to one of the Secretaries. 

12. That nothing be published by the Society which has not been 

previously approved by the Council. 

13. That all questions involving a difference of opinion, be determined 
by a majority of Members at the next meeting. 

14. That if the numbers of the votes be equal, the Chairman have a 

casting vote. 

15. That the annual election of the Officers take place, and the 

accounts of the Treasurer be passed, at the last meeting of the 
Society for the year, in the Easter Term. 

16. That a Special General Meeting may at any time be called by the 

Secretary, in consequence either of instructions received from the 
President, or of a requisition signed by three Ordinary Members. 
The object must be stated to the Secretary, who shall give to each 
Member an intimation thereof, stating, three days previously, the 
time and place of meeting. 

17. That the annual Subscription for each member be One Guinea, to 

be paid in advance, or in lieu of it, a payment of Ten Guineas. 

18. That all persons becoming Members, after the first meeting in 

1820, pay an admission fee of Two Guineas. 

19. That Members be at liberty to introduce each a Visitor; besides 

whom, the President, for the time being, may admit any person, 
with the limitation specified in the succeeding Resolution. 

20. That no resident Member of the University be allowed to 

attend more than two meetings of the Society without becoming 
a Member. 



on resigning office, 27 October, 1890. vii 

At the meeting which took place on the 15th November, these 
draft Regulations were adopted, with some slight changes and 
additions, which are not without interest : 

In Rule 4 the number of ordinary members of the Council was 
increased to seven. To Rule 5, after the words "by ballot" was 
added : " but that the President and Vice-President shall not be 
eligible for more than two years successively ; and that the three 
Senior Ordinary Members of the Council be changed every year." 
To Rule 7 was added : " But that all Noblemen, Heads of Houses, 
Doctors, and Professors, be ballotted for when they are proposed." 
Rule 18 was omitted ; and a new Rule 20 was drawn up: "That 
the Chancellor of the University be requested to accept the office 
of Patron." 

The Regulations, as altered, having been adopted by the meet- 
ing, the Council of the Society for the ensuing year was elected : 

President. 
The Rev. W. Farish, B.D. Magd. Coll., Jacksonian Professor. 

Vice-President. 
John Haviland, M.D. St John's Coll., Regius Professor of Physic. 

Secretaries. 

The Rev. A. Sedgwick, M.A. Trin. Coll., Woodwardian Professor. 
The Rev. S. Lee, M.A. Queens' Coll., Professor of Arabic. 

Treasurer. 
The Rev. B. Bridge, B.D., Fellow of Pet. Coll. 

Ordinary Members of the Council. 

The Rev. E. D. Clarke, LL.D. Jes. Coll., Professor of Mineralogy. 

J. Catton, B.D. Fellow of St John's College. 

T. Turfcon, B.D. Fellow of Catharine Hall. 

T. Kerrich, M.A. Magd. Coll., Principal Librarian. 

R. Wood house, M.A. Fellow of Caius College. 

The Rev. J. Cumming, M.A. Trin. Coll., Professor of Chemistry. 
R. Gwatkin, M.A. Fellow of St John's College. 

We have seen that the meeting at which the above Regulations 
were adopted is called "the first meeting of the Society." The 
Minute Book of the Society, however, takes a different view, and 
places the birthday of the Society a few weeks later. We there 
read : 

" Minutes of the first meeting of the Cambridge Philosophical 
Society held in the Museum of the Botanic Garden, Mon day, December 
13, 1819, Professor Farish in the chair." 



viii Address of Mr J. W. Clark, President, 

At this meeting Professor Farish delivered an address, as also 
did Dr E. D. Clarke. His biographer says : 

" Of this scheme [of founding a Philosophical Society at Cambridge] 
whose direct object was the promotion of science, and its natural 
tendency to raise the credit of the University, Dr Clarke was of course 
one of the earliest and one of the most zealous promoters ; and as it 
was thought advisable, that some address should be provided ex- 
planatory of the design and objects of the Institution, he was requested 
by a sort of temporary council to draw it up. Accordingly he under- 
took the task, and his address having been read at the first meeting, 
was afterwards printed by order of the Society, and circulated with 
the first volume of their Transactions ; although for some reason it 
was not connected with the volume. Nor did his anxiety for the 
support and honour of the Society rest here ; he wrote letters to almost 
all the literary men of his acquaintance, to request their co-operation 
and support; combated with great spirit in several instances, the 
opposition that was made to it from others ; and during the short 
remainder of his life, contributed three papers, which were printed in 
the first volume of their Transactions 1 ." 

Dr Clarke's address is brief, and is chiefly occupied with pointing 
out the advantage of having a society to gather together scientific 
observations which, if scattered through journals, might escape 
notice altogether. It concludes with the following practical sug- 
gestions : 

"Having thus set before the Society the main design and objects 
of its Institution, the Council beg to call the attention of this Meeting 
to considerations of a subordinate nature. It will be necessary to 
provide some place in which the future Meetings may be held, and 
where a repository may be found for the preservation not only of the 
archives and records of the Society, but also of such documents, books, 
and specimens of Natural History, as may hereafter be presented or 
purchased. The utmost economy will at present be requisite in the 
management of the Society's funds ; and therefore if the consent of 
the University could be obtained it would be highly desirable that the 
expenses of printing the Society's Transactions should be defrayed by 
the University. His Royal Highness the Chancellor 2 has accepted 
of the Office of Patron, and his Letter, containing the expression of his 
approbation, will be read by one of the Secretaries. The present Yice- 
Chancellor 3 ; our High Steward 4 ; both our Representatives in Parlia- 
ment 5 ; and many other distinguished Members of the University, 
who are not resident, have also contributed towards the undertaking ; 
and there is therefore every reason to hope that the Graduates of 
this University, who associated for the Institution of the Cambridge 

1 The Life and Remains of the Rev. E. D. Clarke, LL.D. By W. Otter. 4to. 
Lond. 1824, p. 650. 

2 H. E. H. the Duke of Gloucester. 

3 Mr Serjeant Frere, Master of Downing College. 4 Lord Hardwicke. 

5 Viscount Palmerston, M.A. St. John's Coll., and J. H. Smyth, M.A. Trin. Coll. 



on resigning office, 27 October, 1890. ix 

Philosophical Society, by their assiduity and diligence in its support, 
and by their conspicuous zeal for the honour and well-being of the 
University, will prove to other times that their Lives, and their Studies, 
have not been in vain." 

At this meeting the designation of the Society was altered. 
The third Minute runs : 

"That the words 'and Natural History' be added to the second 
regulation, which will then stand as follows, viz. ' That this Society 
be instituted for the purpose of promoting scientific enquiries and of 
facilitating the communication of facts connected with the advancement 
of Philosophy and Natural History.' " 

The change is slight, but not unimportant, for it determined, 
for many years, the direction of the Society's labours. Before long, 
thanks to the enthusiasm and industry of Professor Henslow and 
Mr Leonard Jenyns, it commenced the formation of a Museum, 
long the only Zoological Museum in Cambridge ; and the legitimate 
parent of those collections which I may venture to describe as 
among the most valued possessions of the University. 

The Society was now fairly launched ; the Syndics of the 
University Press undertook to publish the Transactions free of 
charge ; the number of members increased so rapidly that before 
the end of 1820 it had reached 171 ; the finances were in so 
flourishing a condition that £300 was invested in the funds 1 ; and 
opposition gradually died away. " Among the senior members of 
the University," wrote Sedgwick to Herschel, 26 February, 1820, 
" some laugh at us ; others shrug up their shoulders and think our 
whole proceedings subversive of good discipline ; a much larger 
number look on us, as they do on every other external object, with 
philosophic indifference ; and a small number are among our warm 
friends 2 ." 

It was further agreed at the first meeting of the Society: "that 
the High Steward of the University, and the Vice-Chancellor for 
the time being be Vice-Patrons of the Society " ; and at the second 
meeting : "that the members of the Cambridge Philosophical Society 
be designated by the name Fellows of the Cambridge Philosophical 
Society." Early in the following year Dr Clarke proposed : " that 
the Society be hereafter styled The Cambridge Philosophical and 
Literary Society." This proposal was not adopted, as I have 
always thought, most unfortunately. The name would have 
cemented a connexion between science and literature from which 
both would have reaped considerable advantage. As time went on 
the Transactions of the Society would probably have had a literary 
division, as is not uncommon on the continent ; and the first object 

1 Minutes of the Society, 21 February, 1820. 

2 Life of Rev. A. Sedgwick, Vol. i. p. 209. 



x Address of Mr J. W. Clark, President, 

of the Society's formation — the gathering together of observations 
and researches that would otherwise be scattered and lost, would 
have been promoted. 

For a few months the Society met in the lecture-room of the 
building on the east side of what was then the Botanic Garden, 
built in 1784 for the use of the Professor of Botany and the Jack- 
sonian Professor, and now used by the Professor of Pathology. The 
selection of Professor Farish as the first President doubtless deter- 
mined this place of meeting. It was, however, obvious, as Dr Clarke 
had pointed out, that the Society must have a home of its own as 
soon as possible. In April, 1820 1 , arrangements were made for 
securing the use of a house in Sidney Street, opposite to Jesus 
Lane. The Society entered into occupation without delay, and at 
once commenced the formation of a Museum and a Library ; for 
among the Minutes of the first meeting " held in the new rooms," 
1 May, 1820, we find: 

"The thanks of the Society voted to Mr Henslow for his liberal 
donation of a valuable collection in some departments of Natural 
History • and cabinets ordered to be procured for the reception of the 
specimens." 

And at the next meeting (15 May) : 

"The thanks of the Society voted to Dr Clarke, Dr Haviland, and 
Mr Bridge for books presented by them to the Society." 

Again, 13 November, 1820 : 

"The thanks of the Society voted to Mr Henslow for a valuable 
collection of British Insects and Shells systematically arranged in the 
new cabinet." 

The enthusiasm of those days of youth and hope is amusingly 
illustrated by a notice of motion handed in by Dr Clarke : " that 
communications announcing discoveries take the precedence of all 
others." This was agreed to, in a slightly different form, 13 
November, 1820. 

The Society was barely two years old when a project was 
started for giving it a social as well as a scientific side, by establish- 
ing a reading-room, amply stocked with newspapers, reviews, and 
magazines, both English and foreign, as well as with scientific 
journals. A meeting to carry out this scheme was held 22 May, 
1821 ; and so warmly was it taken up that before the end "of the 
year it was agreed that : " the establishment and funds of the 
Reading-room shall be considered as under the control of the 
Society." A committee consisting of the Treasurer (Mr Bridge), 

1 Minutes of the Society, 17 April, 1820. 



on resigning office, 27 October, 1S90. xi 

Mr Oarrighan (Job.), Mr Griffith (Emm.), Mr Peacock (Trim), Mr 
Crawley (Magd.), Mr Whewell (Trin.), Mr Henslow (Joh.), was 
appointed to draw up the following regulations for the manage- 
ment of it, which the Society adopted, 25 March, 1822. 



Rules and Regulations of the Reading Room. 

1. Any Fellow of the Society elected before the 1st of January, 1822, 

may become a member of the Reading-room by writing his name 
in the book for that purpose and paying the subscription of the 
current year. 

2. Every Fellow of the Society elected after the 1st January, 1822, is 

a member of the Reading-room during residence. 

3. Every Fellow of the Society after becoming a member of the 

Reading-room continues so during residence. 

4. Every member of the Reading-room shall pay an annual subscrip- 

tion of one guinea to the Society, the subscription to be due on 
the 1st January for the current year, or he may become a member 
for life by paying ten guineas for the use of the Reading-room in 
any one year. 

5. The following publications shall be taken in 1 . 

6. Every alteration proposed in the list of publications taken in, shall 
be signed by at least three members of the Reading-room, read 
at a meeting of the Society, and suspended in the Room for a 
fortnight during full Term, for any member to signify his assent 
or dissent. If the majority in its favour amount to one-third of 
the signatures and the Council determine that the funds will 
permit, the alteration shall take place. 

7. No Newspaper shall be taken out of the room, and no periodical 

publication shall be removed, before a succeeding number has 
appeared. 

8. Any member, upon taking out a book, shall give to the servant 

of the house, a paper, with the title of the book, signed and dated 
by himself. 

9. Any member violating this rule shall pay a fine of 10s. 

10. Every publication taken out to be returned in a fortnight under a 

penalty of 2s. Qd. 

11. Any member having lost or damaged a book or paper shall replace 
it by a fresh copy of the same. 

1 A space of nearly a full page is left in tlie Minute Book for the list of 
publications, but it has never been written in. 



Xll 



Address of Mr J. W. Clark, President, 



12. The Reading-room shall be open every day from 8 o'clock in the 

morning to 10 at night. 

13. Strangers may be introduced by a member, but no person resident 
in Cambridge can be introduced to the room. 

14. Non-resident Fellows of the Society when visiting Cambridge 

shall be entitled to the use of the Reading-room. 

15. A Steward 1 shall be appointed at the Annual Meeting of the 

Society and considered as a member of the Council. 

16. The office of the Steward shall be : to procure and take care of the 

books, to see that the papers are filed, and the room properly 
prepared for the reception of the members : to collect the bills and 
to sign them before they are paid by the Treasurer. 

It should be remembered that in those days Combination 
Rooms were ill-supplied with newspapers, and the few that 
were taken in were generally in the hands of the Senior 
Fellows. Moreover, in some colleges at least, the juniors were not 
allowed to use the Combination Room at all, except on Feast Days. 
The opportunity therefore, of having access at all times to a well- 
stocked reading-room was eagerly embraced, and formed, with 
many persons, one of the principal inducements to join the Philo- 
sophical Society. 

At the beginning of 1832 it became known that the Society 
would be deprived of the occupation of their house at Midsummer, 
1833 ; nor could another, equally suitable, be either hired or pur- 
chased. Under these circumstances it was decided (7 April, 1832), 
mainly through the influence of Mr Peacock, to apply to St John's 
College for the lease of a site at the corner of All Saints' Passage, 
on which the Society might erect "a house of their own, built 
expressly to suit the objects of the institution." As a preliminary 
to what the Minutes rightly call " this considerable undertaking," 
it was decided to obtain a Charter of Incorporation. The Fellows 
of the Society were evidently warmly in favour of these proposals. 
A sum of three hundred pounds was subscribed in less than a 
month to defray the cost of the Charter ; and at a special general 
meeting held 5 May, 1832, the Council was directed (1) to pre- 
pare a petition for a charter ; (2) to apply to St John's College for 
a building-lease ; (3) "to apply to Mr Humfrey 2 for working-plans, 
and complete estimates for the New House for the Society, the 

1 The Bev. W, Whewell was Steward of the Beading-room from 1822 to 1826, 
when he was succeeded by the Bev. Joh. Lodge, University Librarian. He held 
the office till 1832, when it was discontinued, and a third Secretary was appointed, 
with the understanding that he should have charge of the Beading-room. 

2 A local builder, who obtained the confidence of the University at this period. 
He erected the buildings for Human Anatomy. Architectural History, Willis and 
Clark, Vol. in. p. 156. 



on resigning office, 27 October, 1890. xiii 

same to be submitted hereafter to a general meeting of the Society 
for its approval." Lastly, it was resolved : " that the money re- 
quisite for building the Society's house be raised among the 
members of the Society by shares of £50 each, bearing interest at 
the rate of four per cent, per annum." So eager was the Society 
to begin, that it was decided not to wait for the Charter; the 
plans were approved somewhat hastily, and at a special meeting 
held 16 May, 1832, the architect was directed to invite tenders. 

Early in the Michaelmas Term of 1832 the Charter arrived. 
Professor Sedgwick happened to be President, and, in order to avoid 
additional expense in fees 1 , it had been agreed that his name alone 
should appear upon the document. It therefore begins : "Whereas 
Adam Sedgwick, Clerk, Master of Arts [etc], has by his petition 
humbly represented unto Us, That he, together with others of our 
loyal subjects, Graduates of the said University, did in the year 
One thousand, eight hundred, and nineteen, form themselves into a 
Society," and so forth. No man had a better right to occupy so 
prominent a position ; and it will be readily understood what 
pleasure he himself derived from seeing it there. He was never 
tired of telling the story of the Charter, when, as he put it, "I was 
the Society." 

A special meeting was summoned, 6 November, 1832, to 
accept the Charter. Sedgwick read it, together with an abstract 
of it — and it is almost needless to record that it was accepted 
unanimously. The Council was then directed to prepare a body 
of Bye Laws — the code by which, with only a few slight alterations, 
we are still governed. 

It was on the occasion of the reception of the Charter that the 
first of those dinners was held which have now become an annual 
institution. It seems to me that Sedgwick and the Council of that 
year wished that November 6, 1832, should be kept as the birth- 
day of the Society — to commemorate the fact that on that day it 
had assumed a corporate existence. I need not remind you that 
such a decision involves a sacrifice of twelve years of the Society's 
life ; but, on the other hand, it commemorates an important event 
in its history — for I believe I am right in saying that it was the 
first Society out of London to which a Royal Charter was conceded. 

The new house was ready for the occupation of the Society in 
the autumn of 1833. The situation was convenient, and it was 
itself spacious and well-arranged, with a large meeting-room, 
museum, and reading-room. The change inaugurated an era of 
prosperity which lasted for several years. The meetings were 
well-attended — indeed the Monday evenings on which the Society 
met were held, by common consent, to be pre-occupied, and no 
rival attractions were allowed to interfere with them ; — The 
1 The fees amounted to £271. Minutes, 6 November, 1832. 



xiv Address of Mr J. W. Clark, President, 

museum grew apace, under the fostering care of Professor Henslow 
and his friends ; and the reading-room became more popular than 
ever — a sort of club in fact — where many members of the Uni- 
versity passed several hours of each day, reading and writing or 
conversing with their friends. 

I will next quote an excellent account of the Society's Museum, 
contributed by Mr Leonard Jenyns, in 1838, to The Cambridge 
Portfolio. 

The Cambridge Philosophical Society has been employed from the 
period of its first establishment in 1819, in gradually forming a 
Museum of Natural History. With a view to this end, it has from 
time to time effected several purchases, as well as received the con- 
tributions of various donors. The Museum however is not large ; 
partly owing to the limited funds which can be appropriated to its 
support, and partly to the necessarily restricted space allotted for its 
reception in the Society's house. It is principally, though not exclu- 
sively, devoted to the illustration of the British Fauna. The foundation 
of the Museum may be attributed to Professor Henslow, who presented 
to the Society at its first institution his entire collection of British 
Insects and Shells, arranged respectively in two cabinets. Several 
smaller donations quickly followed, leading the Society to take an 
increased interest in this part of its establishment. In 1828, a spirited 
subscription was commenced amongst its members to assist in pur- 
chasing a most valuable collection of British Birds, for obtaining 
which an opportunity then offered itself. This collection had belonged 
formerly to Mr John Morgan of London. It was extremely rich, 
especially in the rarer species. Many additions however have been 
since made to it ; and the whole forms now a range of thirty large 
cases, which are placed round the principal room in the Museum. The 
birds are beautifully preserved ; and the cases of sufficient size to admit, 
in many instances, of containing entire families. One of the cases 
contains British Quadrupeds. In 1829, the Society purchased a small 
collection of British Insects which was incorporated with that pre- 
viously presented by Professor Henslow. This collection, which con- 
sisted of about 2000 species, was valuable from the specimens having 
been arranged and named by Mr Stephens, the celebrated Entomologist 
of London. Various additions in the same department have been 
since made from time to time by different contributors. In 1833, the 
Society purchased Mr Stephen's entire collection of British Shells, 
contained in two cabinets and comprising a most extensive series of 
species as well as of individuals of each. The Museum has been 
further enriched, in the department of the British Fauna, by a collec- 
tion of Birds' Eggs, presented in part by Mr Yarrell and in part by 
Mr Leadbeater; — also by a collection of Fish, obtained principally on 
the southern shores of the island by Professor Henslow and the 
Rev. L. Jenyns ; — and by a small collection of marine Invertebrate/,, 
obtained at Weymouth by the former of the two gentlemen last 
mentioned. 



on resigning office, 27 October, 1890. xv 

The foreign department of the Museum is not extensive, consisting 
for the most part of single specimens which have been presented at 
different times by different individuals. It contains, however, a small 
collection of reptiles presented by Mr Thomas Bell. It is also rich in 
Ichthyological specimens; having been presented some years back with 
a collection of fish made at Madeira by the Rev. It. T. Lowe ; subse- 
quently, with another collection made in China by the Rev. G. Vachell; 
and yet more recently, with the entire collection of Fish brought 
home from South America and some other portions of the globe by 
C. Darwin, Esq., of Christ's College, and accompanying Naturalist in 
the late voyage of the Beagle, under the command of Captain Fitzroy. 
The whole of the fish above alluded to, as well as those belonging to the 
British collection, are preserved in spirits. They amount to several 
hundred species ; and many of those comprised in the Darwin collection 
are entirely new. Altogether, they constitute a highly valuable as well 
as interesting portion of the Society's Museum. 

Independently of the collections above enumerated, the Philosophical 
Society has made it an object to establish a separate collection of the 
principal animals found in Cambridgeshire. This is a step of the utility 
of which there can be no doubt. Local collections of this nature tend 
to illustrate the Faunas of particular districts; and local Faunas offer 
the best materials for completing our knowledge of the Zoology of the 
whole kingdom. They also throw light upon the geographical distribu- 
tion of animals. In proportion to the number of places in which such 
collections are established, they assist in determining the extreme range 
of the different species, as well as the districts to which they are ordi- 
narily confined. In this department, however, the Birds of Cambridge- 
shire and a few of its Mammalia are alone as yet fitted up for public 
inspection ; but considerable collections have been made in the other 
classes, which are destined one day to take their place in the Museum 
also. 

The Museum of the Society, and that part of it in particular which 
has been just alluded to, has been probably instrumental in exciting 
much interest in the University in the science of Zoology, and diffusing 
amongst its members a taste for such pursuits. Nor is the surrounding 
neighbourhood at all unfavourable for the researches of the naturalist. 
On the contrary, Cambridgeshire may be considered as rich in animal 
productions. From combining within itself a considerable variety of 
soil and situation, it adapts itself to the habits of very different species. 
The fens in particular are inhabited by many rare aquatic birds and 
insects ; and some of these, previous to the introduction of the present 
system of drainage, were in considerable abundance. It may perhaps 
be interesting to mention, that the entire number of vertebrate animals 
found in Cambridgeshire amount to 281. Of these 38 belong to the 
class Mammalia ; 204 to that of Birds ; 9 to that of Reptiles ; and 32 
to that of Fish. The invertebrate animals require further investigation; 
but they probably exceed 9,000, of which the greater portion belong to 
the division of Annulosa. 

The Society has a small collection of minerals and fossils ; but there 



xvi Address of Mr J. W. Clark, President, 

being other Museums in the University devoted to these departments, 
they have received less of its attention than the Zoological part of the 
Museum above noticed. There are also a few antiquities, some of 
which were obtained in the county. 

The Society's house had been built, to a great extent, with 
borrowed money, as I have related, and it had cost a far larger sum 
than had been anticipated. It was possible to pay the interest on 
the loans, but the Society found itself unable to establish a sinking- 
fund for the repayment of the capital. Moreover, the number of 
Fellows gradually decreased. At one time it was usual for nearly 
every Fellow of a College to become a Fellow of the Philosophical 
Society ; but. when the novelty of the existence of such a body 
in Cambridge had worn off, and when the reading-room had 
several rivals, not to mention the reduction of the price of news- 
papers, which enabled them to be taken in at home — there seemed 
to be no special reason for joining a Society where the papers read 
were chiefly mathematical, and which offered no other attractions 
not to be found elsewhere. The officers of the Society did their 
best in these adverse days ; and some of those who had lent money 
cancelled their bonds — as for instance Professor Peacock, Professor 
Sedgwick, Professor Adams, and Professor Babington; but the 
financial difficulty could not be overcome. Finally, in 1865, the 
Museum was offered to, and accepted by, the University 1 ; the 
house was sold ; and the Society found a home at the New 
Museums 2 . 

In this brief review I have of necessity omitted much that 
I should have been glad to record, had I not determined to write 
a sketch and not a history. I cannot, however, conclude without 
drawing attention to our publications. No one, I think, can look 
through the volumes of Transactions and Proceedings without 
admitting that the papers therein printed or abstracted will hold 
their own in originality and value against those of almost any 
society. The Proceedings, as you are aware, do not begin before 
1843. I have therefore appended to this paper brief notices of 
the communications made before that date, as recorded in the 
Minute Book. These will, I feel sure, be found interesting. They 
show what some of the best men in the place were working at ; 
and they testify to the genuine interest taken by them in the 
Society. Whatever they did, they hastened to communicate it, 
though, to our great loss, they too often neglected to prepare their 
work for our Transactions. I have also prepared a list of the 
Presidents, Secretaries, and Treasurers, from the beginning to the 
present time. 

1 Grace, 24 May, 1865. 2 Grace, 8 June, 1865. 



on resigning office, 27 October, 1890. xvii 



OFFICERS OF THE SOCIETY. 

Presidents. 
Date of Election. 

15 Nov. 1819. Rev. Will. Farish, M.A. Magd., Jacksonian Professor. 
22 May, 1821. Rev. Ja. Wood, D.D., Master ofS. John's College. 
13 May, 1823. John Haviland, M.D. Joh., Regius Professor of Physic. 
17 May, 1825. Rev. Ja. Cumming, M.A. Trin., Professor of Chemistry. 
22 May, 1827. Rev. Joh. Kaye, D.D., Master of Christ's Coll. and Bp. 

of Lincoln. 
19 May, 1829. Rev. Tho. Turton, D.D. Cath., Regius Professor of 

Divinity. 
17 May, 1831. Rev. Adam Sedgwick, M.A. Trin., Woodioardian Pro- 
fessor. 
6 Nov. 1833. Rev. Joshua King, M.A. Qu., President of Queens' 

College. 
6 Nov. 1835. Rev. Will. Clark, M.D. Trin., Professor of Anatomy. 
6 Nov. 1837. Rev. Joh. Graham, D.D. Chr., Master of Christ's 

College. 
6 Nov. 1839. Rev. Will. Hodgson, D.D. Pet., Master of Peterhouse. 
6 Nov. 1841. Rev. Geo. Peacock, D.D. Trin., Lowndean Professor. 
6 Nov. 1843. Rev. Will. Whewell, D.D. Trin., Master of Trinity 

College. 
6 Nov. 1815. Rev. Ja. Challis, M.A. Trin., Plumian Professor. 
6 Nov. 1847. Rev. Hen. Philpott, D.D. Cath., Master ofS. Catharine's 

Coll. 
6 Nov. 1849. Rev. Rob. Willis, M.A. Gonv. and Cai. Coll., Jack- 
t sonian Professor. 

6 Nov. 1851. Will. Hopkins, M.A. Pet. 

7 Nov. 1853. Rev. Adam Sedgwick, M.A. Trin., Woodioardian Pro- 

fessor. 
6 Nov. 1855. Geo. Edw. Paget, M.D. Gonv. and Cai. Coll. 
26 Oct. 1857. Will. Hallows Miller, M.D. Joh., Professor of Mine- 
ralogy. 
31 Oct. 1859. Geo. Gabriel Stokes, M.A. Pemb., Lucasian Professor. 
28 Oct. 1861. Joh. Couch Adams, M.A. Pemb., Lowndean Professor. 

26 Oct. 1863. Will. Hepworth Thompson, M.A. Trim, Regius Pro- 

fessor of Greek. 
30 0ot. 1865. Rev. Hen. Wilkinson Cookson, D.D. Pet., Master of 

Peterhouse. 
28 Oct. 1867. Rev. Will. Selwyn, D.D. Joh., Lady Margaret's Pro- 
fessor. 
25 Oct. 1869. Art. Cayley, M.A. Trin., Sadlerian Professor. 
30 Oct. 1871. Geo. Murray Humphry, M.D. Down., Professor of 
Anatomy. 

27 Oct. 1873. Ch. Cardale Babington, M.A. Joh., Professor of 

Botany. 
25 Oct. 1875. Ja. Clerk Maxwell, M.A. Trin., Professor of Experi- 
mental Physics. 

b 



Date of Election. 


29 Oct. 


1877. 


27 Oct. 


1879. 


31 Oct. 


1881. 


30 Oct. 


1882. 


27 Oct. 


1884. 


26 Oct. 


1886. 


30 Jan. 


1888. 



xviii Address of Mr J. W. Clark, President, 

Geo. Downing Liveing, M.A. Joh., Professor of Che- 
mistry. 

Alf. Newton, M.A. Magcl., Professor of Zoology and 
Comparative Anatomy. 

Fra. Maitland Balfour, M.A. Trin. 

Ja. Whitbread Lee Glaisher, M.A. Trin. 

Mich. Foster, M.A. Trin., Professor of Physiology. 

Rev. Coutts Trotter, M.A. Trin. 

Joh. Willis Clark, M.A. Trin. 

Secretaries. 

15 Nov. 1819. Rev. Adam Sedgwick, M.A. Trin., Professor of Geology. 

Rev. Sam. Lee, MA. Qu., Professor of Arabic. 
22 May, 1821. Rev. Geo. Peacock, M.A. Trin. 

Joh. Stevens Henslow, M.A. Joh. 
9 May, 1826. Rev. Joh. Stevens Henslow, M.A. Joh., Professor of' 
Mineralogy and Botany. 
Rev. Will. Whewell, M.A. Trin. 

6 Nov. 1833. Rev. Joh. Stevens Henslow, M.A. Joh., Professor of 

Botany. 
Rev. Will. Whewell, M.A. Trin. 
Rev. Joh. Lodge, M.A. Magd., University Librarian. 

7 Nov. 1836. Rev. Joh. Stevens Henslow, M.A. Joh., Professor of 

Botany. 
Rev. Will. Whewell, M.A. Trin. 
Rev. R. Willis, M.A. Gonv. and Cai. 

6 Nov. 1839. Rev. Will. Whewell, M.A. Trin. 

Rev. R. Willis, M.A. Gonv. and Cai., Jacksonian 

Professor. 
Will. Hopkins, M.A. Pet. 

7 Nov. 1842. Rev. R. Willis, M.A. Gonv. and Cai., Jacksonian 

Professor. 
Will. Hopkins, M.A. Pet. 

Will. Hallows Miller, M.D. Joh., Professor of Mine- 
ralogy. 
6 Nov. 1851. Will. Hallows Miller, M.D. Joh., Professor of Mine- 
ralogy. 
Ch. Cardale Babington, M.A. Joh. 
Geo. Gabriel Stokes, M.A. Pemb., Lucasian Professor. 
6 Nov. 1854. Ch. Cardale Babington, M.A. Joh. 
Joh. Couch Adams, M.A. Pemb. 
Rev. Ch. Fre. Mackenzie, M.A. Gonv. and Cai. 
6 Nov. 1855. Ch. Cardale Babington, M.A. Joh. 
Joh. Couch Adams, M.A. Pemb. 
Geo. Downing Liveing, M.A. Joh. 
25 Oct. 1858. Ch. Cardale Babington, M.A. Joh. 
Geo. Downing Liveing, M.A. Joh. 
Norman Macleod Ferrers, M.A. Gonv. and Cai. 



on resigning office, 27 October, 1890. xix 

Date of Election. 

29 Oct. 1866. Ch. Cardale Babington, MA. Job., Professor of 

Botany. 
Geo. Downing Liveing, M.A. Joh., Professor of Che- 
mistry. 
Rev. T. G. Bonney, M.A. Joh. 
31 Oct. 1870. Rev. T. G. Bonney, M.A. Joh. 
Joh. Willis Clark, M.A. Trin. 
Rev. Coutts Trotter, M.A. Trin. 

27 Oct. 1873. Joh. Willis Clark, M.A. Trin. 

Rev. Coutts Trotter, M.A. Trin. 

Rev. Joh. Batteridge Pearson, B.D. Emm. 

28 Oct. 1878. Joh. Willis Clark, M.A. Trin. 

Rev. Coutts Trotter, M.A. Trin. 

Ja. W hi thread Lee Glaisher, M.A. Trin. 

30 Nov. 1882. Joh. Willis Clark, M.A. Trin. 

Rev. Coutts Trotter, M.A. Trin. 
Will. Mitchinson Hicks, M.A. Joh. 

29 Oct. 1883. Rev. Coutts Trotter, M.A. Trin. 

Ri. Tetley Glazebrook, M.A. Trin. 
Sydney Howard Vines, M.A. Chr. 
26 Oct. 1886. Ri. Tetley Glazebrook, M.A. Trin. 
Sydney Howard Vines, M.A. Chr. 
Jos. Larmor, M.A. Joh. 

31 Oct. 1887. Sydney Howard Vines, M.A. Chr. 

Jos. Larmor, M.A. Joh. 

Matth. Moncrieff Pattison-Muir, M.A. Gonv. and Cai. 
29 Oct. 1888. Jos. Larmor, M.A. Joh. 

Matth. Moncrieff Pattison-Muir, M.A. Gonv. and Cai. 

Sidney Fre. Harmer, M.A. King's. 
28 Oct. 1889. Jos. Larmor, M.A. Joh. 

Sidney Fre. Harmer, M.A. King's. 

Andr. Russell Forsyth, M.A. Trin. 

Treasurers. 

Rev. Bewick Bridge, B.D. Pet. 

Fre. Thackeray, M.D. Emm. 

Rev. Geo. Peacock, M.A. Trin. 

Geo. Edw. Paget, M.D. Gonv. and Cai. 

Rev. Tho. He'dley, M.A. Trin. 

Rev. Will. Magan Campion, M.A. Qu. 

Ja. Whitbread Lee Glaisher, M.A. Trin. 

Rev. Joh. Batteridge Pearson, D.D. Emm. 

Joh. Willis Clark, M.A. Trin. 

Ri. Tetley Glazebrook, M.A. Trin. 



b 2 



15 Nov. 


1819. 


17 May, 


1825. 


6 Nov. 


1834. 


6 Nov. 


1839. 


7 Nov. 


1853. 


26 Oct. 


1857. 


23 Oct. 


1876. 


28 Oct. 


1878. 


29 Oct. 


1883. 


31 Oct. 


1887. 



xx Address of Mr J. W. Clark, President, 



COMMUNICATIONS MADE TO THE SOCIETY. 

February 20, 1820 K 

By Professor Farish (President) : On Isometrical Perspective. 

By Prof. E. D. Clarke : On the discovery of Cadmium in some of the English 

ores of zinc; with some directions respecting the mode of operating. 
By Captain Fairfax (presented hy Mr Okes) : On Soundings at Sea. 

March 6, 1820. 

By Joh. Hailstone, M.A. (Trin.) : On the probable origin of a fossil body found 

on the coast of Scarborough. 
By Professor Farish (President): On Isometrical Perspective (concluded). 

Trans. I. 1 — 19. 
By Joh. Fre. Will. Herschel, M.A. (Joh.) : On functional equations. Trans. I. 

77—87. 
By Mr Okes: On some fossil remains of the Beaver, found near Chatteris. 

Trans. I. 175—177. 

March 20, 1820. 

By Professor Sedgwick : On the Geology of Cornwall, etc. 
By Mr Thompson (Joh.) : A translation from Gemmellaro's account of the 
last great eruption of Etna, in 1819. (Presented by Dr E. D. Clarke.) 

April 17, 1820. 

A letter from the Rev. J. Davis to the Rev. Dr Wood, detailing certain 
optical phenomena observed at Hilkhampton in Cornwall on Wednesday, 
April 5th, 1820, was read to the Society. 

By Joh. Fre. Will. Herschel, M.A. (Joh.) : On the rotation impressed by 
plates of rock crystal on the planes of polarization of the rays of light as 
connected with certain peculiarities in its crystallization. Trans, i. 43 — 
52. 

By Will. Whewell, M.A. (Trin.) : On the position of the apsides of orbits of 
great eccentricity. Trans. I. 179 — 191. 

May 1, 1820. 

By Professor Farish (President) : On the mode of conducting Polar navigation. 
By Joh. Fre. Will. Herschel, M.A. (Joh.) : On certain remarkable instances of 

deviation from Newton's scale in the tints developed by crystals with an 

axis of double refraction on exposure to polarized light. Trans. I. 21 — 

41. 
By Ch. Babbage, M.A. (Trin) : On the Calculus of Functions. Trans. I. 63— 

76. 

May 15, 1820. 

By Mr Emmett : Researches into the mathematical principles of chemical 
philosophy. 

By Prof. E. D. Clarke : On the chemical constituents of the purple precipitate 
of Cassius. Trans. I. 53 — 61. 

By Professor Sedgwick : On the physical structure of Cornwall, etc. (con- 
tinued from 23 March). Trans. I. 89 — 146. 

1 The Minute Book says : " Monday, February 21, 1820 "; but in 1820 February 
21 fell on a Tuesday. 



on resigning office, 27 October, 1890. xxi 

By Sam. Hunter Christie, M.A. (Trin.) : On the laws according to which 
masses of iron influence magnetic needles. Trans. I. 147 — 173. 

A letter from the Rev. J. Davis to the Rev. Dr Wood, containing some 
further details respecting certain optical phenomena mentioned in the 
Minutes of the Society's Meeting on the 17th of April. 

November 13, 1820. 

By Prof. E. D. Clarke : On a method of giving to common Paris Plaster casts 

the appearance of polished Rosso Antico. 
By Professor Lee : On certain astronomical tables by Mohammed al Farsi, a 

MS. copy of which exists in the University Library. Trans. I. 249 — 265. 

November 27, 1820. 

By Prof. E. D. Clarke: On the discovery of native natron in Devonshire. 

Trans. I. 193—201. 
By the same : Notice respecting the sarcophagus brought from Egypt by 

Mr Belzoni. 
By Will. Cecil, M.A. (Magd.): On the application of hydrogen gas to produce 

a moving force in machinery, with the description of an engine where the 

moving force is produced on that principle. Trans. I. 217 — 239. 

December 11, 1820. 

A communication from Dr Wavell (Hon. Member), with some observations 
by Dr E. D. Clarke on the decomposition of a quartzose rock, and on the 
formation of natron. 

By Professor Haviland, Vice-President: On some unusual appearances pre- 
sented by the stomach of a man who died of a fever. Trans. I. 287 — 
290. 

By Professor Lee : A demonstration of the properties of parallel lines, by 
Nasir el Din, translated from the Arabic. 

March 5, 1821. 

"Communications from Professor Leslie and Dr Wavell read by Dr E. D. 

Clarke." 
By Fra. Lunn, B.A. (Joh.) : On the analysis of a native phosphate of copper. 

Trans. I. 203—207. 
By Prof. E. D. Clarke : On the crystallization of water. Trans. I. 209—215. 

March 19, 1821. 

A communication (read by Professor Sedgwick) from Mr Ross respecting 

some minerals found at Buralston. 
By Prof. E. D. Clarke : On Arragonite. 

April 2, 1821. 

By Joh. Leslie, Professor of Mathematics in the University of Edinburgh 

(Hon. Member) : On the effect of hydrogen gas on the propagation of 

sound. (Read by Professor Lee.) Trans. I. 267. 
By Professor Cumming : Exhibition of experiments and communication read 

on the effects of the galvanic fluid on the magnetic needle. Trans. I. 

269—279. 
By Professor Sedgwick : On the geology of the Lizard district. 



xxii Address of Mr J. W. Clark, President, 

May 7, 1821. 

By Joh. Fre. Will. Herschel, M.A. (Joh.) : On the refraction of Apophyllite. 

Trans. I. 241—247. 
By Professor Sedgwick : On the geology of the Lizard (concluded). Trans. I. 

291—330. 

May 21, 1821. 

By Professor dimming : On the connexion between galvanism and magnetism. 

Trans. I. 281—286. 
By Will. Cecil, M.A. (Magd.) : On the application of regulators to machinery. 

November 12, 1821. 

The following communication from Dr Brewster was read by Prof. E. D.' 
Clarke : 

" I have examined with great care a specimen of Leelite, and I find it 
to be an irregularly crystallized body, like Hornstone, Flint, and having a 
sort of quaquaversus structure, or one in which the axes of the elementary 
particles are in every possible direction, instead of being parallel, as they 
must be in all regular crystals. The alumina which Leelite contains gives it 
quite a different action upon light from any of the analogous siliceous sub- 
stances ; and I have thus obtained an unerring optical character by which 
Leelite may be distinguished from them with the greatest facility. 

In examining the different kinds of topazes, I have found that the colour- 
less topazes, and the blue topazes of Aberdeenshire, differ not merely from the 
yellow Brazil topazes, but also from one another." 

Signed, D. Brewster. 
By Mr Okes : On a peculiar case of an enlargement of the ureters in a boy of 
eleven years of age. Trans, i. 351 — 358. 

November 26, 1821. 

By Fre. Thackeray, M.D. (Emm.) : On a remarkable instance of organic 

remains found on the turnpike road between Streatham and Wilburton 

in the Isle of Ely. Trans, i. 459. 
By Will. Mandell, B.D. (Qu.) : On an improvement in the common mode of 

procuring potassium. Trans. I. 343 — 345. 
By Will. Whewell, M.A. (Trin.) : On the crystallization of fluor spar. Trans. 

I. 331—342. 
By Joh. Stevens Henslow, M.A. (Joh.) : On the geology of the Isle of 

Anglesea. 

December 10, 1821. 

By Professor Cumming : On a remarkable human calculus in the possession 

of the Society of Trinity College. Trans, i. 347—349. 
By Joh. Stevens Henslow, M.A. (Joh.) : On the geology of the Isle of 

Anglesea (continued). 
By Ch. Babbage, M.A. (Pet.) : On the use of signs in mathematical reasoning. 

(Ptead by Mr Peacock.) Trans, n. 325—377. 

February 25, 1822. 

By Joh. Hailstone, M.A., late Fellow of Trin. Coll., and Woodwardian Pro- 
fessor : Some observations on the weather, accompanied by an extra- 
ordinary depression of the barometer, during the month of December, 
1821. (Read by the Secretary.) Trans. I. 453—458. 



on resigning office, 27 October, 1890. xxiii 

By Joh. Stevens Henslow, M.A. (Joh.) : On the geology of the Isle of Anglesea 
(concluded). Trans. I. 359—452. 

March 11, 1822. 

The President proposed that, in consequence of the death of the Vice- 
President of the Society, Prof. E. D. Clarke, the meeting should be adjourned 
without proceeding to the regular business of the evening. This proposition 
was agreed to unanimously, and the Society adjourned immediately. 

March 25, 1822. 

By Will. Mandell, B.D. (Qu.) : A description of a new self-regulating lamp. 

By Mr H. B. Leeson : A description of a safety apparatus to the hydrostatic 
blowpipe of Tofts, by which it may be converted into an oxyhydrogen 
blowpipe without danger to the operator. (Read by Mr Peacock.) A 
model of the safety apparatus, and of the blowpipe, was exhibited to the 
Society and explained by Mr Leeson. 

By Geo. Biddell Airy, student of Trinity College: On the alteration of the 
focal length of a telescope by a variation of the velocity of light and of 
the observations to which the change may give rise. (Read by Mr 
Peacock.) 

April 22, 1822. 

By David Brewster, LL.D., Honorary Member of this Society: On the differ- 
ence of optical structure between the Brazilian topazes and those of 
Scotland and New Holland. Trans, n. 1 — 9. 

May 6, 1822. 

By Will. Whewell, M.A. (Trin.) : On the rotation of bodies. Trans. n. 11 — 

20. 
By Dav. Brewster (Hon. Member): On the distribution of the colouring 

matter, and on certain peculiarities in the structure and optical properties 

of the Brazilian topaz. Trans, n. 1 — 9. 

May 21, 1822. 

By Professor Sedgwick : On the basaltic dykes in the county of Durham, and 
the great basaltic formation in Teesdale. Trans. II. 21 — 44. 

November 11, 1822. 

By Will. Whewell, M.A. (Trin.) : On the oscillations of a chain suspended 

vertically, and on the oscillations of a weight drawn up uniformly by a 

string. 
By Fra. Gybbon Spilsbury: On a peculiar relation existing between gravity 

and the production of magnetism in galvanic combinations. (Read by 

Mr Henslow.) Trans, n. 77—83. 

November 25, 1822. 

By Geo. Biddell Airy, Scholar of Trin. Coll. : On the construction of achro- 
matic reflecting telescopes with silvered lenses in the place of metallic 
mirrors. Read by Mr Peacock. Trans, n. 105 — 118. 

December 9, 1822. 

By Will. Cecil, M.A. (Magd.) : On an apparatus for grinding telescopic mirrors 
and object-lenses. Trans, n. 85 — 103. 



xxiv Address of Mr J. W. Clark, President, 

February 17, 1823. 
No papers recorded. 

March 3, 1823. 
No papers recorded. 

March 17, 1823. 
No papers recorded. 

April 14, 1823. 

By Will. Whewell, M.A. (Trin.) : On the different methods which have been 
proposed to grind lenses and mirrors by machinery to a parabolic form. 

By Joshua King, M.A. (Qu.) : A new demonstration of the parallelogram of 
forces. Trans. II. 45 — 46. 

By Geo. Peacock, M.A. (Trin.) : On the analytical discoveries of Newton and 
his contemporaries. 

April 28, 1823. 

By Professor Gumming : On the development of electro-magnetism by heat. 
Trans, n. 47 — 76. 

May 12, 1823. 

By Geo. Peacock, M.A. (Trin.) : On the irregular indications of the thermo- 
meter. 

November 10, 1823. 

By Professor Cumming: On rotation produced by electro-magnetism as 

developed by heat. 
A letter was read by Mr Peacock from Will. Joh. Bankes, M.P., on the 
subject of the manuscript on papyrus of the lost book of the Iliad, recently 
discovered by one of his agents in the island of Elephantina in Upper Egypt, 
accompanied by a facsimile made by Mr Salt of the ten first lines of the 
manuscript. 
By Geo. Biddell Airy, B.A. (Trin.) : Explanation of an instrument exhibited to 

the Society, for the purpose of proving by experiment the constancy of 

the ratio of the sines of incidence and refraction in liquids. (Read 

by Mr Peacock.) 
By Joh. Murray, F.S.A. : Some remarks on the temperature of the egg, as 

connected with its physiology. (Read by Mr Peacock.) 
By the same : Experiments and observations on the temperature developed in 

voltaic action, and its unequal distribution. (Read by Mr Peacock.) 

November 24, 1823. 

By Will. Whewell, M.A. (Trin.) : On the expressions for the cosine of the 
angle between two lines and two planes when referred to oblique co- 
ordinates. Trans, n. 197 — 202. 

By Olinthus Gregory, LL.D. : An account of some experiments made in order 
to ascertain the velocity with which sound is transmitted in the 
atmosphere. (Read by Mr Peacock.) Trans, n. 119 — 137. 

December 8, 1823. 

By Professor Cumming: Exhibition of Dobereiser's experiments of the con- 
tortion of platina wire by a stream of hydrogen gas. 

By Will. Cecil, M.A. (Magd.) : Exhibition of a model of an improved ear- 
trumpet. 



on resigning office, 27 October, 1890. xxv 

By Geo. Peacock, M.A. (Trin.) : On the analytical discoveries of Sir I. Newton. 
(Concluded.) Mr Peacock read three unedited letters of Newton to Dr 
Keill on the subject of the controversy on the discovery of the method of 
fluxions. 

March 1, 1824. 

By Mr Okes : Notice of a magnificent collection of fossil bones, found near 

Barnwell, of the Elephant, Rhinoceros, Buffalo, Deer, Horse. 
By Will. Mandell, M.A. (Qu.): A letter of Sir Isaac Newton to Mr Acland 

of Geneva was read. The following is a copy of the letter. 
Vir celeberrime, 

Gratias tibi debeo quam maximas quod schema experimenti quo lux in 
colores primitivos et immutabiles separatur, emendasti, et longe elegantius 
reddidisti quam prius. Sed et me plurimum obligasti quod schema in 
Camina senea incisum et inter imprimendum obtritum refici curasti, ut 
impressio libri elegantior redderetur. Gratias igitur reddo tibi quam 
amplissimos. Quod inventa mea de natura lucis et colorum viris summis 
Domino Cardinali Polignac et Domino Abbati Bignon non displiceant, valde 
gaudeo. Utinam haec vestratibus non minus placerent, quam elegantissimaj 
vestrae et perfectissimge delineata? picturse nostratibus placuerunt. Ut Deus 
te liberet a doloribus capitis et salvum conservet ardentissime precatur 
Servus tuus humillimus et obsequentissimus 

Dabam Londini Isaacus Newton. 

22 Oct. 1722. 

Celeberrimo viro D n0 . Acland. 
By Professor Sedgwick : On the geology of Teesdale. 

March 15, 1824. 

By Will. Mandell, M.A. (Qu.) : Description of a self-regulating lamp. 

By Geo. Biddell Airy, B.A. (Trin.) : On the figure of the planet Saturn. 

Trans, n. 203—216. 
By Professor Sedgwick : On the geology of Teesdale (continued). 

March 29, 1824. 

By Will. Mandell, M.A. (Qu.) : On a means of protecting locks from the 

insertion of skeleton keys. 
By G. Harvey, F.R.S.E., M.G.S.V. : On the fogs of the Polar seas. 
By Professor Sedgwick : On the geology of Teesdale (concluded). Trans. II. 

139—195. 

May 3, 1824. 

By Cha. Babbage, M.A. (Pet.) : On the determination of the general terms 
of a new class of infinite series. (Read by Mr Peacock.) Trans, n. 217— 
225. 

By Geo. Biddell Airy, B.A. (Trin.) : On the construction of a new achromatic 
telescope. 

May 17, 1824. 

By Joh. Hogg, B.A. (Pet.) : On two petrifying springs in the neighbourhood of 
Norton in the County of Durham. (Read by Professor Henslow.) 

By Geo. Biddell Airy, B.A. (Trin.) : On the principle and construction of the 
achromatic eyepieces of telescopes, and on the achromatism of micro- 
scopes. Trans, n. 227 — 252. 



xxvi Address of Mr J. W. Clark, President, 

May 24, 1824. 

By Professor Haviland, President : On the cases of secondary smallpox, and of 
smallpox after vaccination, which have occurred in Cambridge during the 
last year. 

By Professor Farish: On a method of obviating the inconveniences arising 
from the expansion and contraction of the iron in iron bridges. 

November 15, 1824. 

By Professor Cumming : On the use of gold leaf in the detection of galvanism. 
By Will. Whewell, M.A. (Trin.) : On the principles of dynamics. 

November 29, 1824. 
By Professor Cumming : On the history of electro-magnetism. 

December 13, 1824. 

By Professor Farish : On the construction of the cogs of wheels. 

Professor Farish likewise exhibited to the Society the action of wheels in 
the form of involutes of circles upon each other as an illustration of 
the subject of his paper. 

February 21, 1825. 

By Professor Cumming : On the conversion of iron into plumbago by the 

action of sea- water. Trans, n. 441 — 443. 
By Geo. Biddell Airy, B.A. (Trin.) : On a peculiar defect of his eyes producing 

distortion of images, and on the means of correcting it. Trans, n. 267 — 

271. 
By Professor Sedgwick: On the essential distinction between alluvial and 

diluvial deposits. Annals of Philosophy, x. 1825, pp. 18 — 37. 

March 7, 1825. 

By Will. Whewell, M.A. (Trin.) : On a general method of converting rectilineal 

figures into others which are equivalent, such as squares, etc. 
By Professor Sedgwick : On alluvial and diluvial deposits (continued). 

March 21, 1825. 

By Jos. Power, M.A. (Cla.) : A general demonstration of the principle of 
virtual velocities. Trans, n. 273 — 276. 

April 18, 1825. 
By Professor Farish : On the construction of the cogs of wheels (concluded). 

May 2, 1825. 

By Geo. Biddell Airy, B.A. (Trin.) : On the generation of curves by the 
rolling of one curve upon another, and on the formation of the curves of 
the teeth of wheels which may work in each other with perfect uniformity 
of action. Trails, n. 277 — 286. 

By Professor Sedgwick : A portion of a paper on the geology of the Yorkshire 
coast, a section of which was exhibited to the Society. 

By Will. Whewell, M.A. (Trin.): Exhibition of drawings of the appearances 
■ presented by the spokes of wheels in motion when seen through parallel 
bars, and which consist of a series of quadratures. 



on resigning office, 27 October, 1890. xxvii 

May 16, 1825. 

By Ja. Alderson, B.A. (Pemb.) : An account, with measurements, of an 

enormous whale cast upon the coast of Holderness. (Read by Professor 

Cumming.) Trans. II. 253—266. 
By Professor Sedgwick : On the geology of the Yorkshire coast (concluded). 
By Will. Whewell, M.A. (Trin.): On the classification of crystalline forms, 

particularly with reference to the systems of Weiss of Berlin and Mohs of 

Freyberg. 

November 14, 1825. 

By Ri. Wellesley Rothman, B.A. (Trin.) : On the discrepancies between the 
magnetic intensities at different places on the earth's surface, as deter- 
mined by observation, and by a formula partly empirical and partly 
theoretical of Horsteen and Barlow. Trans, ii. 445. 

By Geo. Biddell Airy, B.A. (Trin.) : On the connection of impact and pressure, 
and the explanation of their effects on the same principles. 

By Leonard Jenyns, M.A. (Joh.) : On the ornithology of Cambridgeshire (read 
by Professor Henslow). 

November 28, 1825. 

By Leonard Jenyns, M.A. (Joh.) : On the ornithology of Cambridgeshire (con- 
cluded). Trans, II. 287—324. 

December 12, 1825. 

By Ch. Babbage, M.A. (Pet.) : On the principles of mathematical notation 
(read by Mr Peacock). 

February 13, 1826. 

By Will. Whewell, M.A. (Trin.) : On the notation of crystallography. Trans. II. 

427—439. 
By Professor Farish : Explanation of a method of correcting the errors from 

the near position of a meridian mark. 

February 27, 1826. 

By Will. Woodall, M.A. (Pemb.) : On a method of finding the meridian line. 
By Professor Farish : Supplement to a paper read at the last meeting. 
By Geo. Peacock, M.A. (Trin.) : On Greek arithmetical notation. 

March 13, 1826. 

By Will. Hen. Wayne, M.A. (Pet.) : On beds containing fossil bones inter- 
mixed with clay and gravel. A letter read, with observations, by Profes- 
sor Sedgwick. 

By Geo. Peacock, M.A. (Trin.) : On Greek arithmetic (concluded). 

April 10, 1826. 

By Geo. Peacock, M.A. (Trin.) : On the origin of Arabic Numerals, and the 
date of their introduction into Europe. 

April 24, 1826. 

By Will. Whewell, M.A. (Trin.) : On a new method of Perspective; particularly 
for objects comprehending a large vertical and a small horizontal space. 

By Geo. Peacock, M.A. (Trin.) : On the origin of Arabic Numerals, etc. (con- 
cluded). 



xxviii Address of Mr J. W. Clark, President, 

May 8, 1826. 

By Geo. Biddell Airy, M.A. (Trin.) : Observations on the Mecanique Celeste of 
Laplace, Book III., with some remarks on the objections of Mr Ivory. 
Trans, n. 379—390. 

By Professor Sedgwick : On the Geology of the Isle of Wight. 

November 13, 1826. 

By Professor Sedgwick: Exhibition of a pair of large fossil horns, of some 
species of the genus Bos, found near Walton in Essex. 

By Will. Whewell, M.A. (Trin.) : On the classification of crystalline combina- 
tions, and the canons of derivation by which their laws may be investi- 
gated. Trans. II. 391 — 425. 

November 27, 1826. 

By Geo. Biddell Airy, M.A. (Trin.) : On the motion of a pendulum disturbed 
by any small force, and on the application of this method to the theory of 
escapements. Trans, in. 105 — 128. 

December 11, 1826. 

By Geo. Peacock, M.A. (Trin.) : On the numerals of the South American lan- 
guages. 

After the meeting Professor Airy gave an account of the construction and 
application of the steam-engine in the mines of Cornwall. 

February 26, 1827. 

By Professor Airy : On the mathematical theory of the Rainbow. 
After the meeting Professor Henslow gave an account of the structure of the 
capsules of mosses, illustrated by coloured drawings. 

March 12, 1827. 

By Geo. Peacock, M.A. (Trin.) : On the discoveries recently made on the 
subject of the Egyptian Hieroglyphics. 

March 26, 1827. 

By Professor Henslow : On the specific identity of the Cowslip, Oxlip, and 
Primrose. 

By Will. Whewell, M.A. (Trin.) : Note on the perspective projection of objects 
on a horizontal plane. 

After the meeting Professor Cumming gave an account of the different forms 
of the Galvanometer, and of the discoveries recently made in Electro- 
dynamics. 

April 30, 1827. 

By Will. Sutcliffe, M.A. (Trin.) : On the application of mathematics to Politi- 
cal Economy, and the effects of a partial Tithe. 

By Will. Whewell, M.A, (Trin.) : On the Perspective of Panoramas. 

After the meeting Professor Sedgwick exhibited a large pair of horns of [some 
species of the genus Bos] found near Walton in Essex ; and an Ichthyo- 
saurus, found at Lyme ; on which he offered some observations. 

May 14, 1827. 

By Will. Sutcliffe, M.A. (Trin.) : On the application of mathematics to Politi- 
cal Economy, etc. (concluded). 



on resigning office, 27 October, 1890. xxix 

By Professor Airy : On the defects of the eye-pieces of telescopes. 

After the meeting Professor Sedgwick gave an account of the peculiarities 
of the Coal Strata in the neighbourhood of Whitehaven : and George 
Noakes (set. 7), a boy remarkable for his powers of calculation, was ex- 
amined by several members of the Society. 

May 21, 1827. 

By R. M. Fawcett : On the use of Iodine in cases of Paralysis. 

By Pi'ofessor Airy : On the observation of eye-glasses depending upon their 

spherical figure, and on the periscopic Panorama. Trans. III. 1 — 63. 
After the meeting Mr Peacock gave an account of the discoveries recently 

made in Hieroglyphics. 

November 12, 1827. 

By Tho. Jarrett, B.A. (Cath.) : On Algebraical Notation. Trans, in. 65—103. 
By Will. Whewell, M.A. (Trin.): On the History and Principles of Chemical 

Nomenclature and Notation, with suggestions of some alterations in the 

Notation hitherto in use. 
By Will. Mandell, B.D. (Qu.) : Exhibition of a piece of breccia, supposed to be 

a fragment of a Roman quern or hand-mill, found on the Hills Road. 

November 26, 1827. 

Professor Sedgwick read a letter from Mr Ri. Tho. Lowe, concerning certain 
petrifactions, apparently of. vegetable origin, which are found in the 
Island of Madeira. 

By Professor Henslow : An account of the application of the chloraret of lime 
to the purpose of disinfecting and neutralizing putrid and noxious sub- 
stances. 

December 10, 1827. 

Dr Fre. Thackeray presented a sword of the sword-fish, and read some obser- 
vations on the bones of the head, and especially those which seem to 
belong to its olfactory system. 

By Leonard Jenyns, M.A. (Joh.) : On the monstrous prolongations of teeth, 
etc., which have been observed in different animals, particularly the 
teeth of a rabbit and the bill of a rook which exist in the Collection of 
the Society ; and on the circumstances by which such deformities have 
been observed to be accompanied. 

February 17, 1828. 

By Alex. Ch. Louis D'Arblay, M.A. (Chr.): Remarks on a pamphlet by Messrs 

Swinburne and Tylecot of St John's College, concerning the proofs of the 

Binomial Theorem, and especially that of Euler. 
After the meeting Mr Peacock gave an account of the representations occurring 

in Egyptian monuments of the deities of that country, and of the funeral 

rituals. 

March 3, 1828. 

By Alex. Thomson (Joh.) : On a mode of obtaining exact measures of the 
cranium. 

By Will. Whewell, M.A. (Trin.) : On the different systems of mineralogical 
classification. 

After the meeting Professor Sedgwick gave an account of the geological struc- 
ture of Scotland, as collected from the observations made by himself and 
Mr Murchison during the preceding summer. 



xxx Address of Mr J. W. Clark, President, 

March 17, 1828. 

By Tho. Jarrett, M.A. (Cath.) : On the development of Polynomials. 

By Will. Whewell, MA. (Trin.) : On the different systems of mineralogical 
classification (concluded). 

After the meeting Hen. Coddington, M.A. (Trin.) gave an account of the ex- 
periments on vibrations and nodal lines of Chladni, Savart, on the con- 
struction of organ-pipes, etc. 

April 21, 1828. 

By Will. Whewell, M.A. (Trin.) : On mineralogical nomenclature. 

By Temple Chevallier, M.A. (Pern.) : On certain properties of numbers. 

By Bob. Willis, B.A. (Cai.) : On the pressure of the air between two discs 

when affected by a stream of air passing through a tube perforating one 

of the discs. Trans, ill. 129 — 140. 
After the meeting Mr Willis exhibited various experiments illustrative of the 

laws of pressure described in his memoir. 

May 5, 1828. 

By Tho. Jarrett, M.A. (Cath.) : On the arithmetic of lines. 
By Professor Whewell : On mineralogical classification (concluded). 
After the meeting Professor Haviland gave an account of the nature and use 
of the stethoscope. 

May 19, 1828. 

By Thos. Jarrett, M.A. (Cath.): On two theorems useful in the integration of 

certain functions. 
By Joh. Will. Lubbock, M.A. (Trin.) : On the calculation of Annuities, and 

some theorems in the doctrine of chances. Trans, in. 141 — 154. 

November 10, 1828. 

By Ja. Challis, M.A. (Trin.) : On the Law of Distances applied to the Satel- 
lites. 

After the meeting Professor Whewell gave a lecture on the granite veins of 
Cornwall. 

November 24, 1828. 
By Professor Airy: On the Longitude of Cambridge. Trans, ill. 155 — 170. 
By Rob. Willis, M.A., Gonv. and Cai. Coll. : On the vowel sounds. 
After the meeting Mr Willis exhibited experiments illustrative of his doctrines. 

December 8, 1828. 

By Joh. Warren, M.A. (Jes.) : On the doctrine of impossible quantities, and 
their geometrical representation, and on the proof that every equation of 
n dimensions has n roots. 

By Fre. Thackeray, M.D. (Emm.) : On the case of Ann Carter, a young woman 
at Stapleford, said to be a trance. 

By Ja. Challis, M.A. (Trin.) : On the Law of Distances, etc. (concluded). 
Tram. m. 171—183. 

After the meeting Mr Leonard Jenyns gave an account, illustrated by draw- 
ings, of the comparative anatomy of Birds and Mammals, and of several 
particulars respecting the former Class. 

March 2, 1829. 

By Pierce Morton, B.A. (Trin.) : On the focus of a conic section. Trans, in. 
185—190. 



on resigning office, 27 October, 1890. xxxi 

By Professor Whewell : On the application of mathematical reasoning to cer- 
tain theories of Political Economy. 

After the meeting Professor Whewell gave an account of various contrivances 
employed in dipping needles, and of some suggested improvements. 

March 16, 1829. 

By Professor Whewell: On the application of mathematical reasoning, etc. 

(concluded). Trans. III. 191—230. 
By Rob. Willis, M.A. (Gonv. & Cai.) : On the theory of the sounds of pipes as 

relating to their vowel quality (concluded from 24 Nov. 1828). Trans, in. 

231—268. 
After the meeting Mr Willis exhibited experiments illustrative of the influence 

of the length of the pipe on the vibrations of the reed, and of the different 

ways in which the vowel sounds may be produced. 

March 30, 1829. 

By Ja. Challis, M.A. (Trin.) : Abstract of a memoir on the vibrations of elastic 

fluids. Trans, in. 269—320. 
By Joh. Will. Lubbock, M.A. (Trin.) : On the tables of the chances of life, and 

on the value of annuities. Trans, in. 321 — 341. 
After the meeting Professor Henslow gave an account of the organization and 

classification of ferns, illustrated by drawings. 

May 4, 1829. 

By Professor Whewell : On the mineral ogical systems proposed by Nordenski- 

old, Bernsdorff, Kefersheim, and Naumann. 
After the meeting Mr Leonard Jenyns gave an account of the construction, 

properties, and mode of growth, of feathers. 

May 18, 1829. 

By Will. Hallows Miller, M.A. (Joh.) : On caustics formed by successive re- 
flexions at a spherical surface. 

By Rob. Willis, M.A. (Gonv. & Cai.): On the mechanism of the human voice. 
Trans, iv. 323—352. 

After the meeting Mr Willis exhibited various experiments and models, and 
explained the action of the organs of voice. 

November 16, 1829. 

By Professor Airy : On a correction of the length of a pendulum consisting of 

a wire and ball. Trans, in. 355 — 360. 
By Professor Whewell : On the causes and characters of Pointed Architecture. 
After the meeting Professor Whewell described the kinds of vaulting employed 

in German churches, with their history; illustrating his account with 

models. 

November 30, 1829. 

By Ri. Wellesley Rothman, M.A. (Trin.) : On an observation of a solstice at 

Alexandria recorded by Strabo. Trans, ill. 361 — 363. 
By Professor Whewell : On Pointed Architecture (concluded). 
By Will. Hallows Miller, M.A. (Joh.) : On the forms and angles of certain 

crystals. Trans, ill. 365—367. 
After the meeting Professor Sedgwick gave an account of the geology and 

structure of the Alps, illustrated by a section from the plains of Bavaria 

to those of Trieste. 



xxxii Address of Mr J. W. Clark, President, 

December 14, 1829. 

By Professor Airy: On the mathematical conditions of continued motion. 

Trans. III. 369—372. 
By Ch. Pleydell Neale Wilton, M.A. (Joh.) : On the geology of the shore of 

the Severn in the Parish of Awre in Gloucestershire. 
After the meeting Mr Leonard Jenyns gave an account of the circumstances 

connected with the migration of Birds. 

February 22, 1830. 

By Ja. Challis, M.A. (Trin.) : On the integration of the equations of motion of 

fluids ; and on the application of this to the solution of various problems. 

Trans, in. 383—416. 
By Leonard Jenyns, MA. (Joh.) : On the Natter-Jack of Pennant, with a list 

of Reptiles found in Cambridgeshire. Trans, in. 373—381. 
After the meeting Professor Henslow explained the discoveries of M. Dutro- 

chet on Endosmose and Exosmose. 

March 8, 1830. 

By Ch. Pleydell Neale Wilton, M.A. (Joh.): Account of a visit to Mount 

Wingen, the burning mountain of Australia. 
By Hen. Coddington, M.A. (Trin.) : On the construction of a microscope 

invented by him, which he exhibited to the Society. 
After the meeting Professor Airy gave an account, illustrated by models, of 

the instruments which have been used to measure altitudes : viz. the 

Zenith Sector, the Quadrant, the Refracting Circle, the large Declination 

Circles of Troughton, and the Circles of Reichenbach. 

March 22, 1830. 

By Will. Hallows Miller, M.A. (Joh.): On the measurement of the angles of 
certain crystals which occur in the slags of furnaces. Trans, in. 417 — 420. 

By Hen. Coddington, M.A. (Trin.) : On the advantages of a microscope of a 
new construction. Trans, in. 421 — 428. 

By Hugh Ker Cantrien, B.A. (Trin.'i : On the Calculus of Variations. 

Mr Willis gave an account, illustrated by models and drawings, of the con- 
struction and muscles of the tongue, palate, and pharynx, and of the 
mode in which these operate in the production of vowel and modulated 
sounds. 

April 26, 1830. 

By Leon. Jenyns, M.A. (Joh.) : On the late severe winter. 

By Hen. Coddington, M.A. (Trin.) : On his new-invented microscope. 

By Professor Whewell : On the proof of the first law of motion. 

After the meeting Professor Whewell gave an account of Gothe's objections to 

the Newtonian theoiy of Optics, and of the doctrine proposed by that 

author. 

May 10, 1830. 

By Tho. Chevallier, M.D. : On the anatomy and physiology of the ear. 

After the meeting Professor Gumming explained the construction and use of 
the areometer of Professor Leslie, and its resemblance to the stereometer 
of Captain Say ; and the construction of an instrument for measuring the 
whole quantity of sunshine which operates during any given time. 



on resigning office, 27 October, 1890. xxxiii 

May 24, 1830. 

By Professor Airy : On the peculiar form of the rings produced by a ray circu- 
larly polarized, and on the calculation of the intensity of light belonging 
to this and other cases. 

By Will. Webster Fisher (Down.) : On the appendages to organs as provi- 
sionary to the modifications of the functions. 

By Bob. Murphy, B.A. (Gonv. and Cai.) : On the general properties of definite 
integrals, and on the equation of Biccati. Trans, ill. 429 — 443. 

By Hen. Coddington, M.A. (Trin.) : A further explanation of his microscope. 

After the meeting Mr Willis exhibited and explained an instrument for 
making orthographical projections of objects. 

November 15, 1830. 

By Aug. De Morgan, B.A. (Trin.), Professor of Mathematics in London 
University : On the Equation of Curves of the second degree. Trans, iv. 
71—78. 

By Will. Okes, M.A. (Gonv. and Cai.) : On the Wourali poison used by the 
Maconshi Indians ; a blow -pipe, quiver, and arrows were exhibited. 

By Professor Cumming : A communication from Mr Edwards on a substance 
resembling cannel coal, found in digging a canal near Norwich. 

By Bi. Tho. Lowe, B.A. (Chr.) : On the Natural History of Madeira. 

After the meeting Professor Whewell gave an account of a method of con- 
structing cross vaults without boarded centering, revived and described 
by M. de Lassaulx of Coblentz. 

November 29, 1830. 

By Ri. Tho. Lowe, B.A. (Chr.): On the Natural History of Madeira (con- 
cluded). Trans, iv. 1 — 70. 

By Professor Whewell : Rules for the selection and employment of symbols of 
mathematical quantity. 

After the meeting Mr Leonard Jenyns gave an account, illustrated by draw- 
ings, of the quinary system of Natural History proposed by Mr M c Leay. 

December 13, 1830. 

By Professor Whewell : Rules for the selection, etc. (concluded). 

By Professor Henslow : On the mode of reproduction of the Chara. 

After the meeting Professor Henslow made some observations on tall ferns, 
exhibiting a specimen of a stalk. 

A machine was exhibited invented by Professor Airy for the purpose of ex- 
hibiting the mode of propagation of undulations along a line of particles. 

February 21, 1831. 

By Professor Airy : On the nature of the two rays formed by the double 
refraction of quartz. Trans, iv. 79 — 123. 

After the meeting Professor Airy exhibited a machine for illustrating the 
nature of the undulations supposed in circular polarization ; an instru- 
ment for exhibiting the rings, spirals, etc. produced by double refraction ; 
and an instrument for exhibiting the same phenomena by means of the 
light produced by the combustion of lime in oxygen. 

March 7, 1831. 

By Rob. Murphy, B.A. (Gonv. and Cai.) : On the general solution of equa- 
tions. Trans, iv. 125 — 153. 

VOL. VII. FT. IV. C 



xxxiv Address of Mr J. W. Clark, President, 

After the meeting Mr Willis exhibited a series of experiments on the trans- 
verse and longitudinal vibrations of strings, membranes, and solids, illus- 
trative of the researches of M. Savart. 

March 21, 1831. 

By Will. Hallows Miller, M.A. ( Joh.) : On the elimination of the time from the 

differential equations of the motion of a point, whether affected by a 

resisting medium, or by any disturbing forces. 
By the same : On measurements of the angles of certain artificial crystals. 
After the meeting Mr Willis exhibited and explained a machine constructed 

for the purpose of illustrating the motion of the particles of any medium 

in which undulations are propagated. 

April 18, 1831. 

By Professor Whewell : On the mathematical exposition of some of the leading 
doctrines of Mr Bicardo's Principles of Political Economy and Taxation. 

By Professor Airy : Notice of an apparatus constructed under his direction by 
Mr Dollond, and of the phenomena of elliptically polarized light exhibited 
by means of the apparatus. Trans, iv. 199 — 208. 

After the meeting Professor Henslow exhibited a series of appearances pro- 
duced by two wheels revolving one behind the other. 

May 9, 1831. 

By Ch. Pritchard, B.A. (Joh.) : A method of simplifying the demonstration of 

the two principal theorems respecting the figure of the earth considered 

as heterogeneous. 
By Professor Whewell : On the mathematical exposition, etc. (concluded). 

Trans, iv. 155—198. 
After the meeting Mr Willis exhibited apparatus illustrating the nature of 

sound, and the vibrations which produce it, especially an instrument 

which he calls a Lyophone. 

May 16, 1831. 

By Ja. Francis Stephens : Description of Chiasognathus grantii, a new Luca- 
nideous insect forming the type of an undescribed genus, together with 
some brief remarks upon its structure and affinities. Trans, iv. 209 — 216. 

By Professor Clark : On a monster of the kind called semidouble. Trans, iv. 
219—255. 

After the meeting Mr Willis exhibited Mr Trevelyan's experiment on the 
rocking of a bar of hot brass placed upon a plate of cold lead. 

Mr Leonard Jenyns gave an account of the application of the quinary system 
of Mr M c Leay to the classification of Birds. 

November 14, 1831. 

By Professor Airy : On some new circumstances in the phenomena of 

Newton's rings. Trans, iv. 279—288. 
By Professor Henslow : On a hybrid plant between Digitalis purpurea and D. 

lutea. 
After the meeting Professor Sedgwick gave an account, illustrated by sections, 

of the geological structure of Caernarvonshire. 

November 28, 1831. 

By Leonard Jenyns, M.A. (Joh.): A monograph of the British species of bivalve 
mollusca belonging to the genera Cyclas and Pisidium. Trans, iv, 289 — 
312. 



on resigning office, 27 October, 1890. xxxv 

By Sam. Earnshaw, B.A. (Joh.) : On the integration of the general linear 
differential equation of the nth. order, and the general equation of 
differences with constant coefficients. 

After the meeting Professor Whewell gave an account of the theories of 
evaporation, the use of Daniel's hygrometer, and the formation of clouds. 

December 12, 1831. 

By Professor Cumming : Exhibition of a calculus found in the intestines of a 

horse, with remarks. 
By Professor Henslow : On a hybrid Digitalis (concluded). Trans, iv. 257 — 

278. 
After the meeting Mr C. Jenyns gave an account, illustrated by drawings, of 

the rules of the perspective of shadows, and explained the use of the cen- 

trolinead. 

March 5, 1832. 

By Professor Airy : On a new analyser of light polarized in a peculiar manner. 

Trans, iv. 313—322. 
By Rob. Murphy, B.A. (Gonv. and Cai.) : On an inverse calculus of definite 

integrals. {Trans, iv. 353 — 408.) 
After the meeting Professor Airy exhibited an apparatus illustrative of some 

of the phenomena referred to in his paper. 
Professor Henslow gave a lecture, illustrated by specimens and drawings, on 

the age of trees. 

March 19, 1832. 

By Professor Airy : On the phenomena of Newton's rings when formed 
between two transparent substances of different refractive powers. Trans. 
iv. 409—424. 

By Will. Brett, M.A. (Corp.) : On the phenomena of double stars. 

After the meeting Mr Whewell gave an account, illustrated by diagrams, of 
the forms and course of the cotidal lines according to the causes which 
influence them, and according to the observations made in different places. 

April 2, 1832. 

By J[ohn] P[rentice] Henslow : On the habits of two hybrid pheasants pre- 
sented by him to the Society. 
By B. Bushell : On the anatomy of the same birds. 
By Will. Brett, M.A. (Corp.) : On the theory of stars of variable brightness. 

May 7, 1832. 
By Sir Joh. Ere. Will. Herschel, M.A. (Joh.) : Description of a machine for 

solving equations. Trans, iv. 425 — 440. 
By Will. Holt Yates, M.B. (Joh.) : Account of the magnetic mountain of 

Sipylus near Magnesia. 
After the meeting Professor Sedgwick gave an account, illustrated by maps 

and sections, of the Physical Geography and History of the Fens of 

Cambridgeshire. 

May 21, 1832. 

By Sir Joh. Fre. Will. Herschel, M.A. (Joh.) : Description of a machine, etc. 

(concluded). 
By Rob. Willis, M.A. (Gonv. and Cai.) : On the ventricles of the larynx. 
By Professor Henslow : On a monstrosity of Reseda. Trans, v. 95 — 100. 
After the meeting Mr Willis gave a lecture, illustrated by experiments, on 

various phenomena of sound. 

c2 



xxxvi Address of Mr J. W. Clark, President, 

June 4, 1832. 

By Joh. Hogg, M.A. (Pet.) : On the classical plants of Sicily. 

By Professor Henslow : Exhibition of a drawing representing the construction 
of Reseda, in illustration of his former paper. 

By Professor Clark : Exhibition of a semi-double fostus of a pig, with 
explanation. 

By Professor Cumming : On Mr Faraday's recent discoveries in magneto- 
electricity, with illustrative experiments. 

November 12, 1832. 

By Geo. Green : Mathematical investigations concerning the laws of the 
equilibrium of fluids analogous to the electric fluid ; with other similar 
researches. Trans, v. 1 — 63. 

By Aug. De Morgan, B.A. (Trin.) : On the general equation of surfaces of the 
second degree. Trans, v. 77 — 94. 

After the meeting Professor Henslow gave an account, illustrated by various 
drawings and diagrams, of various observations of Geology and Natural 
History made by him during a residence at Weymouth during a portion 
of the summer. 

November 26, 1832. 

By Rob. Murphy, M.A. (Gonv. and Cai.) : On an elimination between an in- 
definite number of unknown quantities. Trans, v. 65 — 75. 

By Will. Whewell, M.A. (Trin.) : On the architecture of Normandy. 

After the meeting Mr Ch. Brooke (Joh.) gave an account of the history and 
recent improvements in Lithotripsy, illustrated by the exhibition of the 
instruments used, and by several drawings. 

December 10, 1832. 

By Will. Whewell, M.A. (Trin.) : On the architecture of Normandy (continued). 

After the meeting Mr Sims gave an account of the various methods of engine- 
dividing, and of original dividing practised with regard to graduated 
instruments ; and explained particularly the method of original divid- 
ing invented by Mr Troughton, and recently applied by Mr Sims to the 
division of the mural circle of the Observatory. This explanation was 
illustrated by models and apparatus. 

February 25, 1833. 

By Will. Whewell, M.A. (Trin.) : On the architecture of Normandy (continued) 
After the meeting Professor Airy gave an account, illustrated by models and 

diagrams, of his researches concerning the mass of Jupiter by means of 

observations of the Fourth Satellite. 

March 11, 1833. 

By the Marchese Spineto : An examination of the grounds of Sir Isaac 
Newton's system of chronology. 

After the meeting Professor Sedgwick gave an account, illustrated by repre- 
sentations of sections, of the Geology of North Wales. 

March 25, 1833. 

By Jos. Power, M.A. (Trin. Hall) : On the effect of wind on the barometer. 

By Professor Clark : On an unusual situation of the origin of the internal 
mammary artery, with a drawing and explanation. 

After the meeting Professor Henslow gave an account, illustrated by dia- 
grams, of a method of classifying and designating colours, particularly 
with reference to their use in natural-historical descriptions. 



on resigning ojjice, 27 October, 1890. xxxvii 

April 22, 1833. 

By Professor Miller : On lines produced in the spectrum by the vapour of 
Bromine, Iodine, and Euchlorine. Phil. Mag. 1833, i. 381. 

By Will. Whewell, M.A. (Trin.) : On the architecture of Normandy (concluded). 

After the meeting Mr Whewell explained some of the difficulties which had 
attended his researches concerning co-tidal lines. 

May 6, 1833. 

By Mr Millsom : A description of the anatomy of a hybrid animal — a lion- 
tiger (communicated by Dr Haviland). 

By Geo. Green : A memoir on the exterior and interior attractions of ellip- 
soids (communicated by Sir Edw. Tho. French Bromhead, Bart., M.A. 
Gonv. and Cai.). Trans, v. 395—429. 

By the Marchese Spineto : On an insect which appears in the Egyptian 
Hieroglyphics. 

By Professor Airy : On diffraction. Trans, v. 101 — 111. 

May 20, 1833. 

By Will. Hopkins, M.A. (Pet.) : On the position of the nodes of the vibration 

of the air in tubes. Trans, v. 231 — 270. 
Mr Hopkins also exhibited experiments illustrating the interference of the 

vibrations of the air. 

November 11, 1833. 

By Rob. Murphy, M.A. (Gonv. and Cai.) : A second memoir on the inverse 

method of definite integrals. Trans, v. 113 — 148. 
By Professor Airy : An account of various observations made on the Aurora 

Borealis of September 17 and October 12. Phil. Mag. 1833, ii. 461. 

November 25, 1833. 

By Professor Henslow : Observations on a beetle found in a block of maho- 
gany presented to the Society. 

By Ri. Tho. Lowe, M.A. (Chr.) : Description of a molluscous animal of the 
genus Umbrella, with a drawing and remarks. 

By Will. Hopkins, M.A. (Pet.) : On the geology of Derbyshire, illustrated by 
maps and sections. Phil. Mag. 1834, i. 66. 

December 9, 1833. 

By Hen. Moseley, B.A. (J oh.), Professor of Natural Philosophy in King's Coll. 

Lond. : On the general conditions of the equilibrium of a system of 

variable form ; and on the theory of equilibrium, fall, and settlement, of 

the arch. Trans, v. 293—313. 
By Professor Farish : On the appearance of a meteor, or falling star, of great 

splendour, observed by him at a quarter before seven o'clock, on 

September 26 (he being near Magdalene College). 
By Professor Sedgwick : On the geology of Charnwood Forest, illustrated by 

maps and sections. Phil. Mag. 1834, i. 68. 

February 17, 1834. 

By Fra. Lunn, M.A. (J oh.) : On a specimen of Proteus anguimbs, presented 

by him to the Society. 
By Professor Miller : Some optical observations on lines in the vapour of 

Iodine, Bromine, and Perchloride of Chrome. Phil. Mag. 1834, i. 312. 
By Will. Whewell, M.A. (Trin.) : On the nature of the truth of the Laws of 

Motion. Trans, v. 149—172. 



xxxviii Address of Mr J. W. Clark, President, 

March 3, 1834. 

By Ja. Challis, M.A. (Trin.) : On the motion of fluids. Trans, v. 173— 203. 
By Temple Chevallier, B.D. (Cath. H.) : On the polarisation of the light of 

the atmosphere. Phil. Mag. 1834, i. 312. 
By Professor Miller : Notice of experiments on the Perchloride of Chrome. 

March 17, 1834. 

By Jos. Power, M.A. (Trin. Hall) : On the theory of Exosmose and Endosmose. 

Trans, v. 205—229. 
By Professor Henslow : On Braun's speculations concerning the arrangement 

of the scales on fir-cones, with additional remarks. 
By Professor Airy : On the polarisation of light by the sky, and by rough 

bodies. Phil. Mag. 1834, i. 313. 

April 14, 1834. 

By Professor Airy : On the latitude of the Cambridge Observatory, as deter- 
mined by means of the mivral circle. Trans, v. 271 — 281. 

By Will. Whewell, M.A. (Trin.): On Sir J. Herschel's hypothesis concerning 
the absorption of light by coloured media. Phil. Mag. 1834, i. 463. 

April 28, 1834. 

By Professor Miller : On the axes of crystals. Phil. Mag. 1834, i. 463. 
By Sam. Earnshaw, B.A. (Joh.) : On the laws of motion. Ibid. 
After the meeting Mr Willis explained a machine of his construction for 
jointing together the bones of skeletons. 

May 12, 1834. 

By Aug. De Morgan, B.A. (Trin.) : An attempt to shew that the principles of 

the Differential Calculus may be established without assuming the forms 

of any expansion (in a letter to Mr Peacock). 
After the meeting Professor Miller exhibited and explained an instrument for 

taking the specific gravities of bodies. 
[By Kob. Willis, M.A. (Gonv. and Cai.) : Exhibition and explanation of an 

instrument constructed by himself, which he proposes to call an Ortho- 

graph. 
By Will. Webster Fisher, M.B. (Down.) : On the origin of Tubercular 

diseases.] 

November 10, 1834. 

By Ri. Tho. Lowe, M.A. (Chr.) : Descriptions of six new or rare species of fish 

from Madeira, with drawings. Trans, vi. 195 — 201. 
By Will. Whewell, M.A. (Trin.) : Observations of the tides made from June 7 

to June 22, 1834, at the coastguard stations ; with some observations on 

the mode of discussing them. 

November 24, 1834. 

By Professor Airy : On the rings produced by viewing the image of a star 
through an object-glass of circular aperture. Trans, v. 283—291. 

By the same : On the longitude of the Cambridge Observatory, as compared 
with the result of the Trigonometrical Survey. 

By Ri. Stevenson, B.A. (Trin.) : On the establishment of propositions by the 
infinitesimal method combined with the doctrine of projections. 

By Professor Sedgwick : On the geology of Cambridge. Phil. Mag. 1835, i. 74. 



on resigning office, 27 October, 1890. xxxix 

December 8, 1834. 

By Professor Miller : On the position of the optical axes of crystals. Trans, v. 
431—438. 

By Professor Henslow : On the Green Sand at Haslingfield, Barton, etc. 

By the same : On the age of trees, as determined by their size. 

By Professor Airy : On the echo from the open end of a tall chimney. 

[By Professor Cumming : A statement of Melloni's discoveries on the trans- 
mission of heat by radiation.] 

March 2, 1835. 

By Rob. Murphy, M.A. (Gonv. and Cai.) : On the inverse method of Definite 

Integrals. Trans, v. 315—393. 
By Ri. Stevenson, B.A. (Trin.) : On the solution of some problems connected 

with the theory of straight lines and planes by a new symmetrical method 

of coordinates. 
By Will. Hopkins, M.A. (Pet.) : On Physical Geology. Trans, vi. 1— 84. 

March 16, 1835. 

By Will. Webster Fisher, M.B. (Down.) : On the nature, structure, and changes, 
of tubercles, illustrated by coloured drawings. Phil. Mag. 1835, i. 395. 

After the meeting Mr Willis gave an account, illustrated by drawings and 
models, of the progress of Gothic Architecture, and especially of the 
formation of tracery. Ibid. 

March 30, 1835. 

By Aug. de Morgan, B.A. (Trin.) : On the theorem of M. Abel relative to the 
algebraical expression of the roots of equations which are connected by 
the law of periodic functions. 

By Will. Whewell, M.A. (Trin.) : Exhibition and explanation of an Anemome- 
ter of a new construction ; with a statement of the use which might be 
made of observations made by means of it. 

May 4, 1835. 

By Professor Airy : An account of results recently obtained at the Observatory 

with respect to : (1) the obliquity of the ecliptic ; (2) the mass of Jupiter ; 

(3) Jupiter's time of rotation. 
By Will. Whewell, M.A. (Trin.) : On the results of the Tidal Observations of 

the Coast Guard of June, 1834 ; and on those intended to be made in June, 

1835. 

May 18, 1835. 

By Archib. Smith (Trin.): A communication containing the eliminations by 
which the equation of the wave surface in Fresnel's theory of undulations 
is determined in a manner more simple than in previous investigations 
of other authors on the same subject (read by Professor Airy). Trans. 
vi. 85—89. 

By Will. Whewell, M.A. (Trin.) : An extract of a letter from Professor Schu- 
macher, stating that Messrs Beer and Modler had found the time of 
Jupiter's rotation to be 9 h 55 m 26 s , 5; and that M. Bessul had made a 
long series of observations which give the mass of Jupiter nearly identical 
with those of Professor Airy. 

By Will. Webster Fisher, M.B. (Down.) : On tubercles (continued). 

June 1, 1835. 

By Rob. Willis, M.A. (Gonv. and Cai.) : An account, illustrated by models, of 
the progress of decorative construction in vaults. Phil. Mag. 1835, i. 71. 



xl Address of Mr J. W. Clark, President, 

November 16, 1835. 

By Rob. Murphy, M.A. (Gonv. and Cai.): On the resolution of equations of 

finite differences. Trans, vi. 91—106. 
Extracts of letters written by Sir J. F. W. Herschel, M.A. (Joh.) from the 

Cape of Good Hope, on meteorological observations made by him there. 
Extracts of letters from Ch. Rob. Darwin, B.A. (Chr.) containing accounts of 

the geology of certain parts of the Andes and S. America. 

November 30, 1835. 

By Will. Wallace, F.R.S., Prof, of Mathematics, Edinburgh (Hon. Member): 

On a geodetical problem. Trans, vi. 107 — 140. 
By Professor Airy : On a supposed analysis of the spectrum by Sir D. 

Brewster. 

December 14, 1835. 

By Ri. Potter (Qu.) : On the explanation of the phenomena of the rainbow 

by the doctrine of interferences. Trans. VI. 141 — 152. 
By Ch. Rob. Darwin, B.A. (Chr.) : On viviparous lizards, and on red snow. 

February 21, 1836. 

By Phil. Kelland, B.A. (Qu.) : On the dispersion of Light on the Undulatory 

Theory. Trans. VI. 153 — 184. 
By Will. Whewell, M.A. (Trin.) : On the Tides. Phil. Mag. 1836, i. 430. 

March 7, 1836. 

By Will. Whewell, M.A. (Trin.) : On the recent discoveries of Professor Forbes 
and others respecting the polarisation of heat. Phil. Mag. 1836, i. 430. 

After the meeting Mr Willis gave a lecture on the composition and resolution 
of the entablature in Egyptian and Grecian architecture. Ibid. 

March 21, 1836. 

By Sam. Earnshaw, M.A. (Joh.) : On the solution of the equation of con- 
tinuants of fluids in motion. Trans, vi. 203 — 233. 

By Professor Miller : On the position of the axes of optical elasticity of certain 
crystals. Trans, vn. 209 — 215. 

By Tho. Webster, M.A. (Trin.): On the connection of the periodical [motions] 
of the barometer with the changes of temperature ; and on the relation 
of the accidental changes with the occasional changes. 

April 18, 1836. 

By Professor Sedgwick : An account of the system of formations inferior to 
the Carboniferous Series, as illustrated by his own researches in Wales, 
and those of Mr Murchison in the same country. 

May 2, 1836. 

By Geo. Biddell Airy, M.A. (Trin.), Astronomer Royal : On the intensity of 
light in the neighbourhood of the caustic. Trans, vi. 379 — 402. 

By Will. Hopkins, M.A. (Pet.) : On the agreement between his theoretical 
views of the elevatory geological forces, and the phenomena of faults, as 
observed by him in the strata of Derbyshire. 

May 16, 1836. 

By Aug. De Morgan, B.A. (Trin.) : Sketch of a method of introducing discon- 
tinuous constants into the arithmetical expressions for infinite series. 
(In a letter to Mr Peacock.) Trans, vi. 185—193. 



on resigning office, 27 October, 1890. xli 

By Phil. Kelland, B.A. (Qu.) : On the constitution of the atmosphere and the 

connexion of light and heat. Trans, vi. 235 — 288. 
By Will. Hopkins, M.A. (Pet.): Observations on the temperature of mines, 

and the doctrine of central heat. 
By Geo. Biddell Airy, M.A. (Trin.) : Observations of temperature during the 

great Solar Eclipse of 15 May. 

November 14, 1836. 
No papers recorded. 

November 28, 1836. 

By Joh. Thompson Exley (J oh.): On the leading features of a new system of 
Physics. 

By Professor Henslow: On various kinds of pebbles and agates, with conjec- 
tures respecting the origin of the bands of colour with which they are 
marked. 

December 12, 1836. 

By Sam. Earnshaw, M.A. (Joh.) : On the appearance of light received on a 
screen after passing through an equilateral triangle placed behind the 
object-glass of" a telescope. Trans, vi. 431 — 442. 

By Ja. Jos. Sylvester (Joh.) : On elimination, and the use of indeterminate 
constants. 

By Will. Hopkins, M.A. (Pet.) : On the formation of veins in Derbyshire. 

February 13, 1837. 

By Professor Challis : On the temperature of the higher regions of the atmo- 
sphere. Trans, vi. 443 — 455. 

By Steph. Pet. Rigaud, Savilian Professor of Astronomy, Oxford : On the rela- 
tive proportions of Land and Water. Trans, vi. 289 — 300. 

By Phil. Kelland, B.A. (Qu.) : On the transmission of light through crystal- 
lised media. Trans. VI. 323—352. 

February 27, 1837. 

By Joh. Warren, M.A. (Jes.) : On the algebraical sign of the perpendicular 

from a given point upon a given line. 
By Oh. Rob. Darwin, B.A. (Chr.) : An account of fused sand-tubes found near 

the Rio Plata — which were exhibited, along with several other specimens 

of rocks. 
By Will. Webster Fisher, M.B. (Down.) : On a case of Spina bifida. Phil. 

Mag. 1837, i. 316. 

March 13, 1837. 

By Phil. Kelland, B.A. (Qu.) : Supplement to his paper read 13 February. 

By Sam. Earnshaw, M.A. (Joh.) : On the laws of fluid motion. 

By Hen. Joh. Hales Bond, M.D. (Corp.): A medical-statistical Report of 

Addenbrooke's Hospital for 1836. Trans, vi. 361—377. 
By Will. Whewell, M.A. (Trin.) : An account of the recent results of his 

researches respecting the Tides. 

April 17, 1837. 

By Leonard Jenyns, M.A. (Joh.) : On the temperature of the month of March 

last past. Phil. Mag. 1837, i. 485. 
By Rob. Willis, M.A. (Gonv. and Cai.) : Exhibition and explanation of a 

Tabuloscriptive Engine. Ibid. 



xlii Address of Mr J. W. Clark, President, 



1, 1837. 

By Art. Aug. Moore (Trin.) : Solution of a difficulty in the analysis of 

Lagrange noticed by Sir W. Hamilton (read by Mr Peacock). Trans, vi. 

317—322. 
By Will. Whewell, M.A. (Trin.) : On the results of his Anemometer for the 

first three months of 1837. Trans, vi. 301 — 315. 
By Phil. Kelland, B.A. (Qu.) : On the elasticity of the aether in crystals. 

Trans, vi. 353—360. 

May 15, 1837. 

By Geo. Green (Gonv. and Cai.) : On the propagation of an undulation in heavy 

fluids in a canal of small depth and width. Trans, vi. 457 — 462. 
By Will. Hopkins, M.A. (Pet.) : On the refrigeration of the earth, and on the 

doctrine of internal fluidity of the earth. 
By Hen. Moseley, M.A. (Joh.) : On the theory of the equilibrium of bodies in 

contact. Ti'ans. vi. 463 — 491. 
[By Will. Webster Fisher, M.B. (Down.) : On Spina bifida. Phil. Mag. 1837, 

i. 486.] 

November 13, 1837. 

By Car. Jeffreys, M.A. (Joh.) : Exhibition and explanation of the Respirator 
invented by his brother. 

By Professor Sedgwick : On the geology of Charnwood Forest and the neigh- 
bouring coalfields. 

November 27, 1837. 

By Geo. Green (Gonv. and Cai.) : On the vibration of air. Trans, vi. 403 — 

413. 
By Will. Hopkins, M.A. (Pet.): On certain elementary principles of geological 
theory ; and on Professor Babbage's speculations. 

December 11, 1837. 

By Geo. Green (Gonv. and Cai.) : On the reflexion and refraction of light in 

non-crystallised media. Trans, vn. 1 — 24. 
By Ri. Wellesley Rothman, M.A. (Trin.) : On the observation of Halley's 

comet in 1836. Trans, vi. 493—506. 
By Will. Hopkins, M.A. (Pet.) : On Precession and Nutation, assuming the' 

interior fluidity of the earth. 

February 26, 1838. 

By Dav. Tho. Ansted, B.A. (Jes.) : On a new genus of fossil shells. Trans, vi. 

415— 422. 
By Aug. De Morgan, B.A. (Trin.) : On a question in the theory of probabilities. 

Trans, vi. 423—430. 
By Dan. Cresswell, D.D. (Trin.) : On the squaring of the circle. 

March 12, 1838. 

By Phil. Kelland, M.A. (Qu.) : On molecular attraction. Trans, vn. 25 — 59. 
By Professor Henslow : On plants brought by Mr Darwin from Keeling 
Island. 

March 26, 1838. 

By Geo. Biddell Airy, M.A. (Trin.) : On the intensity of light in the neighbour- 
hood of a caustic. Trans, vi. 379- — 402. 
By Professor Challis : On the proper motions of the stars. 



on resigning office, 27 October, 1890. xliii 

April 30, 1838. 

By Ri. Potter, B.A. (Qu.) : On a new correction in the construction of the 

double achromatic object-glass. Trans, vi. 553 — 564. 
By Will. Hen. Trentham, M.A. (Joh.) : On the expansion of a polynomial. 
By Hen. Joh. Hayles Bond, M.D. (Corp.): Statistical report on Addenbrooke's 

Hospital for 1837. Trans, vi. 565—575. 
By Pet. Bellinger Brodie, B.A. (Trin.) : On the occurrence of recent land and 

fresh-water shells with bones of some extinct animals in the gravel near 

Cambridge ; communicated by Professor Sedgwick. Trans, viii. 138 — 

140. 

May 14, 1838. 

By Joh. Tozer, B.A. (Gonv. and Cai.) : On the application of mathematics to 
calculate the effects of the use of machinery on the wealth of a com- 
munity. Trans, vi. 507 — 522. 

By Duncan Farquharson Gregory, B.A. (Trin.) : On the real nature of 
symbolical algebra. 

By Professor Miller : On measures of spurious rainbows. 

May 28, 1838. 

By Professor Miller : An account of experiments illustrating the unequal 

expansion of crystals by heat. 
By Ri. Tho. Lowe, M.A. (Chr.). On the Botany of Madeira 1 . Trans, vi. 523 

—551. 

November 12, 1838. 
By Professor Whewell : On certain rude kinds of architecture. 

November 26, 1838. 

By Duncan Farquharson Gregory, B.A. (Trin.) : On the logarithms of 

negative quantities. 
By Professor Henslow : On the formation of mineral veins, illustrated by a 

specimen. 

December 10, 1838. 

By Hamnett Holditch, M.A. (Gonv. and Cai.) : On rolling curves (com- 
municated by Professor Willis). Trans, vn. 61 — 86. 

By Ri. Wellesley Rothman, M.A. (Trin.) : On the climate of Italy. 

By Ri. Tho. Lowe, M.A. (Chr.) : An additional note on the Flora of Madeira. 

By Professor Henslow: On the structure of wasps' nests, illustrated by 
specimens. 

February 18, 1839. 

By Ri. Wellesley Rothman, M.A. (Trin.) : On the climate of Italy (concluded). 
By Ri. Potter, B.A. (Qu.) : On the determination of the value of (X), the 

length of an undulation of light. 
By Geo. Green, B.A. (Gonv. and Cai.) : Appendix to a former paper on waves, 

read 15 May, 1837. Trans, vn. 87—9'. 

March 4, 1839. 

By Will. Hopkins, M.A. (Pet.) : On the geology of England and France in the 
neighbourhood of the Channel. 

1 This paper is not mentioned in the Minutes. The date assigned to it is derived from the 
Transactions. 



xliv Address of Mr J. W. Clark, President, 

March 18, 1839. 

By Sam. Earnskaw, M.A. (Joh.) : On the equilibrium of a system of particles. 

Trans, vn. 97 — 112. 
By Geo. Biddell Airy, M.A. (Triu.) : On the diurnal changes of the variation 

of the magnetic needle. 

April 22, 1839. 

By Duncan Farquharson Gregory, B.A. (Trin.) : On photogenic drawings. 
By Professor Sedgwick : On the geology of Cornwall and Devon. 

May 6, 1839. 

By Professor Miller : On the calculation of halos, according to Fraunhofer's 

theory. 
By Geo. Green : Note on reflection and refraction. Trans, vn. 113 — 120. 
By Duncan Farquharson Gregory, B.A. (Trin.) : On chemical classification. 

May 20, 1839. 

By Geo. Green, B.A. (Gonv. and Cai.) : On the motion of light through crys- 
tallised media. Trans, vn. 121 — 140. 

By Dav. Tho. Ansted, B.A. (Jes.) : On the tertiary formations of Switzerland. 
Trans, vn. 141 — 152. 

By Professor Whewell : An account of observations made with his Anemo- 
meter since May, 1837. 

November 11, 1839. 

By Professor Whewell : On a new theory of the Tides. Phil. Mag. 1839, 
ii. 476. 

November 25, 1839. 

By Professor Sedgwick : On the geology of northern Germany, east and west 
of the Bhine. 

December 9, 1839. 

By Aug. De Morgan, B.A. (Trin.) : On the foundations of Algebra. Trans. 

vn. 173—187. 
By Dav. Tho. Ansted, B.A. (Jes.) : On the geology of the Transition Rocks in 

the north-east of Bavaria and the Principality of Reuss. 
By Will. Webster Fisher, M.B. (Down.) : On the malformation of certain 

parts of the nervous system. 

March 2, 1840. 

By Geo. Biddell Airy, M.A. (Trin.) : On a new construction of the Going 
Fusee, applied in the Northumberland telescope. Trans, vn. 217 — 
277. 

By Ch. Pritchard, M.A. (Joh.) : On the achromatism of the telescope. 

March 16, 1840. 

By Aug. De Morgan, B.A. (Trin.) : On the foundation of algebra. 
By Joh. Tozer, M.A. (Gonv. and Cai.) : On some doctrines of Political 
Economy. Trans, vn. 189 — 196. 

March 30, 1840. 

By Phil. Kelland, M.A. (Qu.) : On the quantity of light intercepted by a 
grating placed before a lens. Trans, vn. 153 — 171. 



on resigning office, 27 October, 1890. xlv 

May 4, 1840. 
By Phil. Kelland, M.A. (Qu.) : On the Law of molecular attraction. 

May 18, 1840. 

By Dav. Tho. Ansted, M.A. (Jes.): On the Green Sandstone formation of 

Blackdown, Devon. 
By Professor Miller : On the structure of the Heliotropes of Gauss, Steinheil, 

and Schumacher. 

June 1, 1840. 

By Will. Hopkins, M.A. (Pet.) : On certain geological phenomena of elevation, 
and their connection with the formation of volcanoes. Phil. Mag. 1840, 
ii. 154. 

November 16, 1840. 
By Roderick Impey Murchison : On the geology of Russia. 
By Geo. Biddell Airy, M.A. (Trin.) : On an optical fact, and its explanation 
on the undulatory theory. 

November 30, 1840. 
By Professor Henslow : On the diseases of wheat. 

December 14, 1840. 

By Aug. De Morgan, B.A. (Trin.) : On the composition of forces. 
By Professor Whewell : On the equilibrium of oblique arches. 

February 22, 1841. 

By Professor Whewell : Additional remarks on oblique arches. 
By the same : Is all matter heavy 1 Trans, vn. 197 — 207. 

March 8, 1841. 

By Joh. Tozer, M.A. (Gonv. and Cai.) : On some mathematical formula) for 
determining the permanent effects of emigration and immigration on 
numbers. Phil. Mag. 1841, i. 318. 

March 22, 1841. 

By Professor Miller : On supernumerary rainbows. Trans, vn. 277 — 286. 

April 26, 1841. 

By Professor Challis : On the resistance of air to a pendulum with a spherical 
bob. Trans, vn. 333—353. 

May 10, 1841. 

By Professor Willis : On the arrangement of the joints of crustaceous 

animals. 
By the same : On the original nomenclature of Gothic mouldings. 

May 24, 1841. 
By Professor Challis : On a new kind of interference of light. 

November 15, 1841. 

By Professor Sedgwick : An account of the comparative classification of the 
older strata in the British Isles, 



xlvi Address of Mr J. W. Clark, President, 

November 29, 1841. 

By Aug. De Morgan, M.A. (Trin.) : On the foundation of algebra. Trans. 

VII. 287—300. 
By Jos. Power, M.A. (Trin. Hall) : On the late accident, on the Brighton 

railway. Trans, vn. 301 — 317. 
By Joh. Fre. Stanford, B.A. (Chr.) : On a newly invented locomotive. 

December 13, 1841. 

By Will. Hopkins, M.A. (Pet.) : On the forms of the isothermal surfaces within 
the earth ; and on the thickness of the earth's solid crust, supposing the 
central portion to be fluid. 

By Aug. De Morgan, B.A. (Trin.) : On the foundation of algebra (continued). 

February 14, 1842. 

By Bob. Leslie Ellis, M.A. (Trin.) : On the foundations of the doctrine of 
chances. Trans, viii. 1 — 6. 

February 28, 1842. 

By Bob. Leslie Ellis, M.A. (Trin.) : On the doctrine of chances (concluded). 
Trans, viii. 1 — 6. 

March 14, 1842. 

By Mr Taplin : On the solution of a cubic equation. 

By Professor Whewell (Master of Trinity College) : Are cause and effect 
simultaneous or successive? Trans, vn. 319 — 331. 

April 11, 1842. 

By Professor Challis : On the differential equations of fluid motion. Trans. 

vn. 371—396. 
By Professor Owen : On the fossil remains of a new genus of Saurians called 

Rhynchosaurus, discovered in the New Bed Sandstone of Warwickshire. 

Trans, vn. 355—369. 

April 25, 1842. 

By Matth. O'Brien, M A. (Gonv. and Cai.) : On the propagation of luminous 
waves in the interior of transparent bodies. Trans, vn. 397 — 437. 

By Geo. Gabriel Stokes, B.A. (Pemb.) : On the steady motion of incompressible 
fluids. Trans, vn. 439 — 453. 

May 9, 1842. 

By Professor Kelland : On the motion of glaciers. 

By the same : On the laws of fluid motion. 

By Jos. Power, M.A. (Trin. Hall) : On fluid motion. Trans, vn. 455 — 464. 

By Professor Miller : An account of the Dioptrische Untersuchungen of Gauss. 

November 14, 1842. 

By Professor Fisher : On the development of the spinal ganglia in animals, 
and on the malformation of various portions of the nervous system 
in Man. Phil. Mag. 1842, ii. 485. 

November 28, 1842. 

By Matth. O'Brien, M.A. (Gonv. and Cai.) : On the intensity of reflected and 
refracted light, the absorption of light, and the stability of the luminous 
eether. Trans, viii. 7 — 26, 



on resigning office, 27 October, 1890. xlvii 

December 12, 1842. 
By Will. Hopkins, M.A. (Pet.) : On the glaciers of the Bernese Alps. 

February 20, 1843. 

By Art. Cayley, B.A. (Trin.) : On some properties of determinants. Trans. 

viii. 75—88. 
By Matth. O'Brien, M.A. (Gonv. and Cai.) : On the absorption of light by 

transparent media. Trans, viii. 27 — 30. 

March 6, 1843. 

By Professor Challis : On a new general equation in Hydro-dynamics. 
Tram. viii. 31—43. 

By Geo. Kemp, M.B. (Pet.) : On the nature of the biliary secretion : to shew- 
that the bile is essentially composed of an electro-negative body, in 
chemical combination with one or more inorganic bases. Trans, viii. 
44—49. 

March 20, 1843. 

By Professor Sedgwick : On Professor Owen's memoir on the skeleton of the 
Mylodon ; and on the structure and habits of certain extinct genera of 
gigantic Sloths. 

May 1, 1843. 

By Professor Challis : On the comet of 1843. 

By Will. Williamson, M.A. (Cla.) : Two letters on the same subject. 
By Will. Hopkins, M.A. (Pet.) : On the motion of glaciers. Trans, viii. 
50—74. 

May 15, 1843. 

By Hamnett Holditch, M.A. (Gonv. and Cai.) : On small finite oscillations. 

Trans, viii. 89—104. 
By Professor Willis : On the vaults of the Middle Ages. 

May 29, 1843. 

By Geo. Kemp, M.B. (Pet.) : On the relation between organic and organized 
bodies ; with some remarks on the theory of organic combinations as 
proposed by Laurent. 

By Geo. Gabriel Stokes, B.A. (Pemb.) : On some cases of fluid motion. 
Trans, viii. 105—137. 

October 30, 1843. 

By Will. Hopkins, M.A. (Pet.) : An account of the large reflecting telescope 
which the Earl of Bosse is now constructing ; with an account of the 
manner in which its 6 feet speculum has been prepared. 

November 13, 1843. 

By Professor Sedgwick : An account of the structure and relations of the 
slate rocks of North Wales. 



Between 1831 and 1843 the Proceedings of the Society were reported — 
somewhat irregularly — in the Philosophical Magazine. The notices, as a 
general rule, are extremely brief ; but I have thought it worth while to add 
references to those papers that are not printed in the Society's Transactions 



xlviii Address of Mr J. W. Clark, President. 

whenever the abstracts give details. Moreover, this Journal preserves the 
titles, and brief abstracts, of four papers not noticed in the Minutes of the 
Society. These I have included between square brackets. They belong to 
the meetings held May 12, December 8, 1834 ; May 15, 1837. 



P. S. Since writing the above sketch, I have discovered that a somewhat 
similar scheme for the establishment of a Philosophical Society had been 
projected in 1782. " The death of some persons interested in the plan, and 
several accidents, occasioned the scheme to be postponed till February 18th, 
1784," when Professor Milner, Mr Farish, and some others "associated them- 
selves under certain laws and regulations." They were presently joined by 
several of the most distinguished men in the University, among whom occurs 
the illustrious name of Porson, and a volume of " Tracts, Philosophical and 
Literary, by a society of gentlemen of the University of Cambridge" was 
projected, but never published, though two of the contributions were printed. 
"This little society of learned men, not being adequately supported, was 
dissolved about the close of the year 1786 1 ." One promoter at least of this 
noble, though unsuccessful, attempt, Mr Farish, Jacksonian Professor from 
1813 to 1837, became a member of the Philosophical Society. 

i This information is derived from : Memoirs of John Martyn, F.R.S., and of Thomas Martyn, B.D., 
F.R.S., F.L.S., Professors of Botany in the University of Cambridge. By G. C. Gorham. 8vo. Lond. 
and Camb. 1830, p. 165. 



INDEX OF CONTRIBUTORS. 



A. 

Airy, Geo. Biddell, xxiii, xxiv, xxv, xxvi, 
xxvii, xxviii, xxix, xxx, xxxi, xxxii, 
xxxiii, xxxiv, xxxv, xxxvi, xxxvii, 
xxxviii, xxxix, xl, xli, xlii, xliv, xlv 

Alderson, Ja., xxvii 

Ansted, Dav. Tho., xlii, xliv, xlv 

B. 

Babbage, Ch., xx, xxii, xxv, xxvii 
Bankes, Will. Job.., xxiv 
Bond, Hen. Joh. Hales, xli, xliii 
Brett, Will., xxxv 
Brewster, Dav., xxii, xxiii 
Brodie, Pet. Bellinger, xliii 
Brooke, Ch., xxxvi 
Bushell, B., xxxv 

C. 

Cantrien, Hugh Ker, xxxii 

Cayley, Art., xlvii 

Cecil, Will., xxi, xxii, xxiii, xxiv 

Challis, Ja., xxx, xxxi, xxxii, xxxviii, 

xli, xlii, xlv, xlvi, xlvii 
Chevallier, Temple, xxx, xxxviii 

,, Tho., xxxii 

Christie, Sam. Hunter, xxi 
Clark, Prof., xxxiv, xxxvi 
Clarke, Prof., xx, xxi, xxii, xxiii 
Coddington, Hen., xxx, xxxii, xxxiii 
Cresswell, Dan., xlii 
Cumming, Prof., xxi, xxii, xxiv, xxvi, 

xxviii, xxxii, xxxiii, xxxv, xxxvi, xxxix 

D. 

D'Arblay, Alex. Ch. Louis, xxix 
Darwin, Ch. Bob., xl, xli 
Davis, J., xx, xxi 

De Morgan, Aug., xxxiii, xxxvi, xxxviii, 
xxxix, xl, xlii, xliv, xlv, xlvi 

E. 
Earnshaw, Sam., xxxv, xxxviii, xl, xli, 
xliv 



Ellis, Bob. Leslie, xlvi 

Emmett, Mr, xx 

Exley, Joh. Thompson, xli 

F. 

Fairfax, Capt., xx 

Farish, Prof., xx, xxvi, xxvii, xxxvii 
Fawcett, B. M., xxix. 
Fisher, W. W., xxxiii, xxxix, xli, xlii, 
xliv, xlvi 

G. 

Green, Geo., xxxvi, xxxvii, xlii, xliii, 

xliv 
Gregory, Duncan Farquharson, xliii, 

xliv 
,, Olinthus, xxiv 



Hailstone, Joh., xx. xxii 

Harvey, G., xxv 

Haviland, Prof., xxi, xxvi, xxx 

Henslow, Joh. Stevens, xxii, xxiii, 

xxviii, xxix, xxxi, xxxii, xxxiii, xxxiv, 

xxxv, xxxvi, xxxvii, xxxviii, xxxix, xli, 

xlii, xliii, xlv 
,, Joh. Prentice, xxxv 

Herschel, Joh. Fre. Will., xx, xxii, xxxv, 

xl 
Hogg, Joh., xxv, xxxvi 
Holditch, Hamnett, xliii, xlvii 
Hopkins, Will., xxxvii, xxxix, xl, xli, 

xlii, xliii, xlv, xlvi, xlvii 



J. 

Jarrett, Tho., xxix, xxx 
Jeffreys, Ch., xlii. 
Jenyns, C, xxxv 

,, Leon., xxvii, xxix, xxx, xxxi, 
xxxii, xxxiii, xxxiv, xli 



Index of Contributors. 



K. 
Kelland, Phil., xl, xli, xlii, xliv, xlv, xlvi 
Kemp, Geo., xlvii 
King, Josh., xxiv 



Lee, Prof., xxi 

Leeson, H. B., xxiii 

Leslie, Prof., xxi 

Lowe, Ei. Tho., xxix, xxxiii, xxxvii, 

xxxviii, xliii 
Lubbock, Joh. Will., xxx, xxxi 
Lunn, Fra., xxi, xxxi, xxxvii 



M. 
Mandell, Will., xxii, xxiii, xxv, xxix 
Miller, Will. Hallows, xxxi, xxxii, 

xxxiv, xxxvii, xxxviii, xxxix, xl, xliii, 

xliv, xlv, xlvi 
Millsom, Mr, xxxvii 
Moore, Art. Aug., xlii 
Morton, Pierce, xxx 
Moseley, Hen., xxxvii, xlii 
Murchison, Eoderick Impey, xlv 
Murphy, Bob., xxxiii, xxxv, xxxvi, 

xxxvii, xxxix, xl 
Murray, Joh., xxiv 



N. 
Newton, Sir I. , xxv 

0. 
O'Brien, Matth., xlvi, xlvii 
Okes, Will., xx, xxii, xxv, xxxiii 
Owen, Bic, xlvi 



Peacock, Geo., xxiv, xxv, xxvii, xxviii, 

xxix 
Potter, Bi., xl, xliii 
Power, Jos., xxvi, xxxvi, xxxviii, xlvi 
Pritchard, Ch., xxxiv, xliv 

E. 

Bigaud, Steph. Pet., xli 
Boss, Mr, xxi 



Eothman, Bi. Wellesley, xxvii, xxxi, 
xlii, xliii 

S. 

Schumacher, Prof., xxxix 

Sedgwick, Prof., xx, xxi, xxii, xxiii, 
xxv, xxvi, xxvii, xxviii, xxix, xxxi, 
xxxiv, xxxv, xxxvi, xxxvii, xxxviii, 
xl, xlii, xliv, xlv, xlvii 

Sims, Mr, xxxvi 

Smith, Archib., xxxix 

Spilsbury, Fra. Gybbon, xxiii 

Spineto, Marchese, xxxvi, xxxvii 

Stanford, Joh. Fre., xlvi 

Stephens, Ja. Fra., xxxiv 

Stevenson, Ei., xxxviii, xxxix 

Stokes, Geo. Gabr., xlvi, xlvii 

Sutcliffe, Will., xxviii 

Sylvester, Ja. Jos., xli 



Taplin, Mr, xlvi 

Thackeray, Fre., xxii, xxix, xxx 

Thompson, Mr, xx 

Thomson, Alex., xxix 

Tozer, Joh., xliii, xliv, xlv 

Trentham, Will. Hen., xliii 

W. 

Wallace, Will., xl 

Warren, Joh., xxx, xli 

Wavell, Dr, xxi 

Wayne, Will. Hen., xxvii 

Webster, Tho., xl 

Whewell, Will., xx, xxii, xxiii, xxiv, xxvi, 
xxvii, xxviii, xxix, xxx, xxxi, xxxii, 
xxxiii, xxxiv, xxxv, xxxvi, xxxvii, 
xxxviii, xxxix, xl, xli, xlii, xliii, xliv, 
xlv, xlvi 

Williamson, Will., xlvii 

Willis, Bob., xxx, xxxi, xxxii, xxxiii, 
xxxiv, xxxv, xxxviii, xxxix, xl, xli, xlv, 
xlvii 

Wilton, Ch. Pleydell Neale, xxxii 

Woodall, Will., xxvii 

Y. 

Yates, Will. Holt, xxxv 



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